Does any of these networks work in the uk?

You can buy an iphone 5 in USA for $199 but only available on AT&T, sprint, Verizon contracts and I live in the UK so do any of them work in England

They all do but the phone is $199 when you sign a 2 year contract with Sprint, Verizon or AT&T so when you use it in England you will be roaming internationally which is VERY expensive.

Yes they do, an iPhone? I'd think about getting.that.

No their all American company's
You have to use a uk company because you will be charged roaming charges
try O2, giffgaff, 3, Vodafone, EE

Define "work"
to bring a US contract phone to the UK,
yes iphone4/4S and 5 will work in the UK and Europe on their original SIM 'roaming'
but not on a local SIM unless you get it unlocked

to buy it and use on a UK network:
an iPhone locks to the 1st network SIM that is put in it.
(unless it is Verizon, who have said theirs will be sold unlocked)

So an iPhone bought on most networks will be locked until the network lets you unlock it - and most do not until your 2 years minimum contract ends.

yes, just like you can get an iPhone 'for free' on a 2 year contract in the UK and other countries, the price you see advertised is the contract price.

The sim free/unlocked price (no contract) is here:
http://store.apple.com/...ly/iphone5 - not the headline price
the one at the bottom where is says 'Or get iPhone unlocked and contract-free'
and add Sales tax to that also (another 10% or so)

Without proof of US residence, you will not get a US contract (to run away from).