I use Longbows extensively, so I can tell you what's effective against them.
-- They are strong offensively, but weak defensively. The problem is that range, which lets them pick off approaching attackers. Mounted units are obvious for their speed. If you have Lancers, this may be a good time for them.
-- Also try to use terrain against Longbows. They can't shoot over obstacles. They aren't highly maneuverable. In hilly terrain, crossbows are just as good as longbows, so you can even the score. Try to make them attack you through terrain of your choosing.
-- If you are going to attack a longbow, try to make sure you kill it. The last thing you need is longbows with extra bonuses. Don't know how smart the AI is on this, but my 4th bonus for Longbows is always the second attack. Lethal. So go after units you can kill, not just injure, as much as possible.
-- If your opponent goes after one of your cities with Longbows, that advice about killing them is especially important. But also remember that massive army of just longbows can't take out a city. In desperate straits, focus your counterattacks on their mounted and melee units.
-- Plan your attacks so that you expose fewer units to fewer longbows during their turn. Give them a defensively strong melee unit to shoot at, especially one that you can lose without slowing down your attack. I recommend not throwing cannons at them for this reason -- just too slow to move and set up for the shot. Hold your cannons off until you're doing what cannons were meant to do -- besieging their cities.
-- And on that note, remember you are not at war with the Longbows. You are trying to capture London. If it's a coastal city, use whatever you've got (Galeass/Frigate bombardment and Caravel/Privateer raids) to tag it. This is important because as good as mounted units are against longbows, they suck against cities. If London is not coastal, then you need to plan carefully, but focus on bringing it to submission as quickly as possible.
-- Another possibility is that you would be wasting too much time to go after London at this point in the game and might be better off hitting it with Rifles, Cavalry, Artillery, Landships, Ironclads, etc. AFTER you take out another opponent. Don't assume that because it's closest it's the best opponent for you to take on right now. Longbows aren't effective against more advanced units. They do upgrade to extra range Gatlings, which are fairly hardy, but no longer have that +3 edge that was so devastating in the early eras. This is when more conventional strategies can take them out, even if they are a fairly strong opponent.
-- If you do decide to wait, be prepared for their next UU if water is a factor at all. Ships of the Line are nasty and can get extra range and extra attack bonuses pretty easily. Not sure whether AI are smart enough to go for those, but if you're fighting England in a Renaissance/Industrial era match, you'll need to adapt your naval strategy accordingly.
Good luck! I'm increasingly convinced that England is one of the toughest civilizations in G&K.
