Re: Are you a US citizen who moved to the UK? How did you do

What you want to do is not easy. You can try to find an employer who would sponsor a work permit for you, but that depends on difficult to find skills. Being a nurse would certainly help.

Coming as a tourist is an option, but you should be very careful not to overstay your visa. Otherwise you will be flagged as an illegal immigrant, and getting into the UK will get very difficult.

You should also consider other European countries. Especially the new members of the EU may be looking for certain skills that are not in demand in the UK.