I have a French work visa and might be commuting to the UK for work (US Citizen) every week. Am I going to have to get a UK visa as well? I know they are not Schengen and everything is complicated for the UK.
On a long-stay visa, you have the right to travel in other EU countries (as a tourist or short-term visitor) pretty much on the same basis as if you were coming over directly from the US.
If you're going to be in the UK for a period that exceeds what you'd be able to do on your US passport, then you probably need to look into visa requirements. But if you're only going to be in the UK a couple of days a week, you should be good for six months or a year. (The limit is six months total out of a year.)