You may well be better off getting a bike over in the States, and simply packing it up. I know for sure British Airways and Virgin Atlantic will ship a bike for free at this time (obviously check the fine details on the respective web sites). Even if you have to pay if you use another airline, it's probably still worth it; bikes in the UK aren't cheap. I've cycle toured in the UK a couple of times, both times brought my bikes.
Another reason for doing this? You really don't want to be over there to find out the bike isn't quite right for you or you haven't had the time to get used to it (that is a big deal, otherwise you could have a miserably uncomfortable tour!). Of course you have to balance this against the chance of your bike being damaged or lost in transit, but the same can happen to a bike being shipped to you in the UK anyway! Also, another thing to factor in is if you choose to buy a bike in the UK, what if you need to go a distance to pick the bike up? Travel also isn't cheap in the UK. Just a few thoughts.