US citizens working in the UK are normally 'sponsored' by their employer to do a job here, it does not grant residency , the sponsorship ends when the job finishes. (UK citizens can also be 'sponsored' to work in the USA for an employer.)
They can not come over here and look for work, their company must have a UK job for them and they are sent to do that job.
I used to work for the UK branch of Xerox Corporation which is based in New York, all the senior managers were US citizens who were seconded (sponsored) to work in the UK. Same applies to other US companies with branches in the UK.