I am a US citizen residing in the UK, working for one company under a
UK Work Permit. I will be switching jobs in three weeks' time. My
current work permit is still valid for another 2 years, but it is tied
to my current employer. I understand the new employer can take this
permit over, but needs to apply to get that done. What is the formal
process? Who can help me through it? How long does the process take?
What happens if there is a lapse while the transfer is being
processed? Cost?

Why not start by asking the American Embassy?

Even if they don't know, they can surely point you in the right direction.

What if the person or I want to end the employment?
 Neither of you need our permission to end the employment but you
should inform us in writing if this happens. Please note, however,
that if you or the person end the employment for which the work permit
was issued, the person will cease to meet the requirements of the
Immigration Rules under which they have stayed in the UK and unless
they have another legal basis of stay in the UK they will be expected
to leave. Even if the person has a period of leave to remain left in
their passport, once they have left the employment for which a work
permit was issued they, and any dependants, cannot stay in the UK on
this basis. It is important therefore, that the person seeks
appropriate further leave to remain before the end of the employment
if they want to stay in the UK. While Work Permits (UK) will consider
a work permit application for a change of employment in such
circumstances, a person does not have permission to remain in the UK
beyond the original employment to look for new employment or to
continue in supplementary employment.
http://www.workpermits.gov.uk/default.asp?PageId=1730

For general information/introduction see
http://www.workpermits.gov.uk/default.asp?PageId=1558

Contact information is at
http://www.workpermits.gov.uk/default.asp?PageId=3018
Note this is by alphabetical name of employER.

There is employers information at
http://www.workpermits.gov.uk/default.asp?pageid=17

Work permit application forms can be downloaded from
http://www.workpermits.gov.uk/default.asp?PageId=1576

Owain

Thanks Owain. That's quite useful.