The post above gives the right outline. Just a few details:
1). It is not just British citizens that have advantage in the current UK job market - it is citizens and permanent residents of ALL EU states. By law, the employers cannot even ask the EU citizens to prove their knowledge of English language - much less any medical credentials (so I have worked alongside an Italian nurse with pretty rough English and a Polish doc who had no idea how to manage a sickle cell crisis because he had never seen one in his life back in Poland).
2). Up until very recently (last year), the US-trained psychiatrists could quite easily have had their US credentials recognised in the UK. In fact, I heard there was a US-trained attending who did a 3-year-long stint at the Norfolk and Norwich hospital in East Anglia just for the heck of it (that was about 5 years ago, so no more details - sorry!). Unfortunately, with PMETB came many changes so currently my understanding is that you have to go through a rather complex and convoluted process to become eligible to practice as a Consultant (Attending) in the NHS. It is not impossible, just more difficult than it used to be. Having said that, who knows what they are going to do in 4 years time, when you are finished with your US residency - the Brits seem to like drop profound and unexpected changes on the heads of their unsuspecting citizens. Google MTAS and you will see for yourself.
3). Despite #1, do not despair - if you make enough money and have an advanced degree, you still could be eligible for a job in the UK through the Tier 1 General Migrant scheme (formerly known as Highly Skilled Migrant Programme). Just google it.
4). Be prepared to spend some money and money on licensure process - a friend who is going through a similar process right now tells me it costs over 1K GBP and has already taken a couple of months.
5). I would not say that core specialist training is like basic residency and higher specialist training in the UK is like fellowship training in the US. The key difference is that you can practice as an attending in Adult Psych after 4 years of residency in the US, but you still need 3 years of higher specialist training (so, 3+3 altogether) to become a consultant in the UK. The higher specialist training is NOT optional, unlike a fellowship.
6). You probably will not get a job in sought-after areas (Norfolk and Norwich area is looked down upon by locals, btw), but who knows...If you are a hot shot academic psychiatrist, you might get a place somewhere more competitive. But then, what do you want to get out of your time in the UK? Someone I know wants to spend some time in Wales (where Brits are practically bribed to come into by the UK government); that attending I referred to above (N&N) wanted some authentic English village experience - if that is what you are after, you probably will not be disappointed.
Feel free to PM me with more specific qs.