You should be able to get a US phone unlocked without any problems.
However, here are a few points you might also want to consider:
1. All phone manufacturers modify the Android operating system in some way. (I'm assuming that you're not seeking an iPhone or a Windows phone). They typically include their own choice of default apps, which take up memory but can't be removed. Any phone destined for the US market is likely to have apps on it which are designed specifically for use in the States (such as for US weather services, for connecting to Amazon USA, etc). You could end up with a phone cluttered up with things you can neither use nor remove ;
2. The charger which comes with a US phone will be designed to work from a 120V, 60Hz mains supply, with a US plug. You might need to buy a new charger to get it to work on the UK's 240V, 50Hz supply ;
3. You'll probably get hit for 20% VAT (plus possibly a small amount of Import Tax) when you come through UK customs ;
4. You won't be able to test the phone before you leave the USA. (Mobile phones don't work in the USA until they've been registered with a service provider. Service providers are required to be sure of the ID of someone before giving them access to their services) ;
5. If the phone goes wrong you'll have great difficulty making a claim against the retailer. A manufacturer might refuse to honour any warranty on what they'd see as a 'grey import'.