Republic is not a warmongering government
Republic can do anything it wants. I jump into it ASAP, then go conquer the world (unless I take my time and get so absolutely bloated that switching to Communism is worth the time). Yeah, it's bad for being at war forever, but I've never been in a war when WW became such a problem that I *needed* to make peace.
EDIT:
Okay, have just looked at the game. First ...
You founded Entremont (your capital) in 3850. Why?
In no particular order ...
You need more workers. Ten isn't anywhere near enough, especially with how spread out you are.
You have all the VCs enabled - if you're going for a Conquest victory, why not just have that enabled?
You have a very uneven city placement - a city could fit between Entremont and Cataractonium very nicely - and it's very far as well, especially your first two cities, neither of which were touching Ent's Culture boundary when they were founded.
You don't need the Cathedral you're building. You don't need the Walls or the Barracks.
You *do* need more Marketplaces - you're a Republic, cash is your big advantage. With Roads on every worked tile, Marketplaces, and (if needed) Courthouses, you should have a massive income.
At this point, you don't need the Forbidden Palace. You don't have enough Cities to make it worthwhile.
You have a Leader in Entremont. What is he doing? Either make an Army with him or use him to rush something.
More Culture wouldn't hurt - enemies have nearly-open shots deeper into your territory because it's unowned.
You have too much empty space - if you can't get more Culture to fill in the gaps, plant a city there.
You have extraneous units - there are three in Entremont, Agedincum, Cataractonium, Ratae Coritanorum, and Eboracum. You don't get Happiness bonuses for having units in cities as a Republic, so pull some out and go wail harder on China. Further, you have a lot of unupgraded units - either upgrade them or disband them and build higher-tier replacements.
Finally, you halted your expansion ... you could have greatly increased your number of cities if you'd kept pumping out Settlers to take land that was unoccupied.
Edit2: If you find yourself running out of time, increase the number of turns to 1000. I play Huge Pangaea maps with 1000 Turns, and find myself either winning or being in a completely unassailable position by 400 turns in.