Basically, ranking up your options:
Use your US phone in roaming. This is most convenient but (unless you use T-Mobile and are on their Simple Choice plan with free sms/3g data in roaming) most expensive. Assuming your phone and provider supports it, this way you don't have to do anything at all.
Buy an "international" SIM card in US. This is less convenient (extra hassle), but also likely less expensive than using your phone in roaming. There are plenty of providers with various coverage, and UK is likely covered by most of them - check with each provider though. Also beware of various hidden costs and fees (connection fee, activation fee, daily/monthly fee, inactive balance fee) which some providers like to add up. So far in every case I tried that it assigned you the US number, so this was essentially the special case of option #1.
Buy SIM card in UK. This is the least convenient (you'd have to spend time), but you'd get the cheapest prices from all three options (again, unless you're on that TMO plan). You also would get the local number, which makes difference if you need to receive calls from locals.
Rent the phone at your destination. This is popular option in countries like Korea, where getting a SIM card for a tourist without the residence card doesn't seem to be possible. This is even bigger hassle as you need to line up both to get the phone at the airport AND to return it, and the price is not cheap. This is however the only option if your current phone is locked, and you don't want to purchase another phone.
Of course options #2 and #3 would require your phone to a) be unlocked and b) support SIM cards.
So, as you see, it costs less than roaming, but generally not less than a local prepaid plan. Again, beware of hidden costs - and it is indeed extra hassle to make sure you know all costs in advance. Whether this makes it good time investment or not, only you can decide.