About 200,000 expats in London. There's a booming need for immigration to the US, US wills, purchase of US real estate, and emigration from the US, including giving up citizenship and greencards.
Most potential clients will want you to have some dual US/UK experience. Clauses that are common in US wills may cause problems with their UK situation, and certain legal terms have different meanings and different treatment over here. Greencards carry a heavy potential exit tax burden, and people will be more impressed with you if you push other kinds of visas instead. Alas, your background being solely US will not be what the expats need; there are dozens of lawyers here already who have dual experience and they will be more likely to turn to them instead.
However, there are not enough lawyers who understand enough UK law to be of service to the entire community, so if you want to gain the experience necessary to assist this market sector, come on over and get your background experience.
Try White & Case or Clifford Chance. You may have to take a paycut. Try applying directly from here, rather than via the US offices. The cost of a local hire is lower for them than the cost of an expat hire; so you're more likely to get a favourable response by applying locally than by applying from the US.
Many UK law firms are happy to take on a US lawyer as an adjunct to their practice. You would probably be a part-time consulting lawyer for them, rather than a direct employee. If, when you get here, you don't have a job yet, go and visit dozens of UK law firms and network, network, network. Go to every law and legal event possible and make connections.
Given all of the above, I'd think twice before you decide to set up a solo practice.
When you get here, give me a ring and I'll toss a few bones to you.