Hardly. Britain entered the war because of its absolute guarantee of Belgian neutrality. The Home Rule Bill was only shelved after the war began, and it was Asquith who did so, when he essentially already had it in the bag. In fact, the Home Rule Bill passed and hit the statute books in September 1914, but because of the suspension, did not take effect until after the war - which they anticipated would last only a few months. Asquith doing so was a political scandal of sorts in the complete opposite direction from what you suggest: a political truce on the issue was agreed upon by Unionists and Nationalists at the start of the war, which Asquith deliberately abused his power as majority coalition leader to get around.
So while it was Asquith who led the country to war (Britain had been secretly cooperating with France since the 1905 crisis to coordinate just how a BEF would operate on French soil), doing so was directly in contrast to two core things: 1. his pet project the Government of Ireland Act, which he was basically required to do as part of his winning over the Irish MPs to his coalition, and 2. his Liberal membership, a decidedly peace-oriented party even in response to summer 1914 events. For him to go to war in order to postpone his own project, and against the politics of his party and constituency, would be utter madness. And I don't think Asquith was a hatmaker.
The Belgian issue is and always was a red herring. If the British had intervened under the terms of the Treaty of London, they would have had to call a four power conference in order to sanction any sort of intervention - Metternich, Talleyrand, and the other men who orchestrated the treaty specifically included those terms in order to prevent a power such as France from abusing them by claiming spurious violations of neutrality and using them as a vehicle to spread influence. The British, obviously, had no intention of actually doing this in 1914, because it would be
prima facie ridiculous and would have wasted time anyway. Grey's ultimatum to Germany - rightly - barely mentioned Belgium.
Furthermore, the claim that the British were acting on behalf of Belgian neutrality falls apart when looked at their own war planning and actions. Had the French been the country to invade Belgium first, not the Germans - something that was theoretically possible given Joffre's contingency planning and his positioning of the Fifth Army, that is, if the Germans had simply decided to rewrite their war plans from scratch - the British absolutely would not have intervened against
them. British war plans had been tied to those of the French for nearly ten years by the time the First World War started, and they were predicated not on a German invasion of Belgium but on a state of war existing between Germany and France. Belgium was irrelevant; the Franco-German war was everything. Indeed, the British themselves planned to violate both Belgian and Dutch neutrality before the full report of German movements came in, by landing troops to seize Antwerp and attack the operational flank of the German forces, perhaps even to attack the Rhineland and the heart of German industry. (This move was suggested by Churchill at a sort of ad hoc war council, and generally approved of; it formed the basis for Churchill's later political showboating during the siege of Antwerp in the fall.)
This whole kerfluffle over Belgian neutrality has led to the tiresome tradition of painting the Entente powers, especially the British, of being protective of the rights of small nations or some similar garbage. Because, you know, the illegal British blockade that basically wiped out the Dutch and Danish trading economies was totally instituted to respect those small nations. As was the Anglo-French occupation of neutral Greece for two years, and the Anglo-French decision to fight a war on neutral Greek soil, and the Anglo-French instigation of a Greek civil war, and the eventual Anglo-French intervention to resolve that civil war. And don't forget the Anglo-Russian conspiracy to carve up the Ottoman Empire by provoking the CUP into declaring war by, among other things, interfacing with nationalist movements inside Ottoman borders in Armenia and Yemen, instituting a blockade of the Ottoman Mediterranean coast, menacingly moving an Indian Army expeditionary force to the doorstep of Iraq, and skirmishing with Ottoman troops in the Sinai. And then there's the Japanese violation of Chinese neutrality to capture Qingdao, and then the Japanese imposition of the Twenty-One Demands, largely ignored by Britain, on neutral China while using the war as an excuse. There's the British and Russian violation of Swedish neutrality to destroy iron shipments with their own submarine warfare (the Russians even violated it on the surface at least once)...I mean, the list just goes on and on. Not that I am venting, or anything.
Where the violation of Belgian neutrality was
truly important was in selling the war to the Cabinet, and to the Liberal back bench. While the Tories were more or less gung-ho for a Continental war, Asquith had serious problems convincing even his own Cabinet to approve of war. He and Grey even outright lied to the German ambassador, Lichnowsky, to prevent any serious discussions with the Germans before the Germans had to invade Belgium and so spark a Cabinet realignment. Had the Germans not invaded Belgium, Asquith may very well have had to deal with a mass resignation. (Lloyd George was on the fence, but at least three other ministers seem to have threatened to resign.) In the event that that occurred, Bonar Law had made contacts, through the traitorous army officer Henry Wilson, that he would be prepared to join with Asquith in a grand coalition to prosecute the war. Asquith himself was convinced of the necessity of this anyway, and gave Grey his full support even after various revelations concerning Grey's secret, non-Cabinet-sanctioned, highly illegal diplomacy over the course of the previous few years with respect to a future Franco-German war. So Britain would still have gone to war with Germany, with Asquith as the PM, had Germany not invaded Belgium, but the price would have been a confusing and serious Cabinet reshuffle, and possibly a delay in the outfitting and provision of the BEF.
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Describing the Government of Ireland Act as an Asquith pet project is overstating the case somewhat, anyway. Asquith himself vacillated between semi-enthusiasm (when everything was going well) and a rather lukewarm feeling (the other 99% of the time) about the Act; it was widely supposed that John Redmond was basically holding the Liberal government hostage over it. The question was less one of wanting to push the Home Rule bill through than of keeping the Liberals securely in power for five years until the next election. What seals it is Asquith's complete refusal to have anything to do with Home Rule until the 1911 elections forced him to rely on Irish Nationalist support on confidence issues. Tory parliamentary tactics between 1911 and 1914 correctly focused on this perceived rift between Liberals and Nationalists, although the Liberal whips were able to keep the party in line for the most part; the Tories also committed a serious error at the same time by threatening civil war over the Ulster issue, which redefined the issue from the Six Counties to one of outright treason. (This error was arguably a necessary one for the Tories to commit in order to keep the party from fracturing even more - it neatly avoided the murderous issue of free trade - and it was almost certainly necessary for Bonar Law to commit it because his hold on the party was so weak after his selection as a compromise candidate.)
Asquith was clearly willing to give up the Nationalists and their pet projects so long as he could retain power, but that isn't even the whole issue. Remember, the British military had just endured a "mutiny" and a major military reshuffle that indicated, at the
very least, that the troops in Ireland were unreliable at best in a shooting war with the Ulster Volunteers. The UVF was well armed due to the British and Irish tradition of gun ownership, and had ample reserves after the success of the Larne gun-running. Shooting had already started at Bachelors Walk, and the perceived last effort at a compromise had collapsed with the demise of the Buckingham Conference in July. To all intents and purposes, it looked to everyone that Britain was teetering on the edge of civil war in late July and early August 1914. If Asquith were able to find a way to ditch Home Rule, he would effectively be saving the country from that civil war. And since everybody except Kitchener and a few officers in the CIGS saw any British involvement in a European war as short-term, purely naval, or both, joining the First World War to prevent a Fourth British Civil War seemed like a no-brainer.
Hell, Churchill - not the greatest source on actual military operations or diplomacy, but just fine and dandy for his insights into how the Cabinet worked while he was in it - basically admitted as much when he wrote
The World Crisis: the European war was a way to put Ireland on ice. "The parishes of Fermanagh and Tyrone faded back into the mists and squalls of Ireland, and a strange light began immediately, but by perceptible gradations, to fall and grow upon the map of Europe."
I have written, at some length, a history article, located in my signature, on the crisis generated by the Government of Ireland Act 1914, and although it's clearly geared towards demonstrating that a real danger of war existed at the time more than on the respective political factors governing the decisions of the Liberals and Conservatives (digression: the issue of demonstrating the danger of war is a more relevant one in current Home Rule historiography, in my opinion, because it seems as though most historians of the period regard the UVF as a well-disciplined force willing to comply lockstep with Bonar Law's, Craig's, and Carson's orders, while at the same time too poorly armed to do much of anything without support from Belfast, let alone London - so if Bonar Law, Craig, and Carson were bluffing, a reasonable supposition, civil war would not have happened...but these assumptions about the UVF are clearly wrong, and those about the Tory leaders' motivations are unclear, end digression), it does, by necessity, touch on those issues.