You can't work while on a tourist visa. Period. No, it doesn't matter that you are working for a US company. You simply cannot do work of any kind while on a tourist visa. So, no it won't work. You also cannot change from a tourist visa to any other kind of visa from within the UK. If you came to the UK on a tourist visa and found an employer to sponsor you (extremely difficult, by the way due to the small fixed number of visas available and the poor economy) you wold have to return to the US to apply for the Tier 2 visa.

 

Just to explain further what nyclon has said. You aren't supposed to job hunt while you are in UK as a visitor, and if the Border Agency finds out or suspects you may be doing it, you will be denied entry. If you are a business visitor, there is a list of permitted activities, none of which allows you to look for another job. Supposing you do manage to get into UK as a visitor, and you are to contact potential employers about jobs, the fact that you need a work visa means 99.9% will simply refuse to consider you at all.

You should find out while you are in the US about any companies willing to sponsor you (by networking and using whatever contact you may have). Software engineering isn't a shortage occupation and any potential employer has to pass residential labour market test by advertsing the post for a month in professional journals etc and showing they have received no application from suitably qualified candidates in UK and EU.