The thing is that without a residency visa or UK passport, you would be subject to visitor visa rules in the UK - and that means that every single time you go over to France or Belgium, you will have to 'reapply' for visitor visa entry back into the UK again... and if the immigration officers think you are spending too much time in the UK and are essentially trying to live here for several months per year, they can refuse you entry. Each time, you would need proof of a return ticket back to the US, enough money to support your visit in the UK without working, and evidence of your job and home back in the US to show you are not trying to live in the UK as a visitor.
Also, as a meteorologist, I can tell you that the summer weather in the UK isn't much to write home about. Most Brits leave the country and head for places like SoCal in order to actually see some sunshine and feel some warm temperatures! UK summer temperatures are more like Palm Springs December temperatures and it rains maybe 1 in 4 days (about 2 inches per month).
Average summer highs are 68-70 degrees and we only get maybe 5-10 days per year of temperatures over 75. If we're really lucky we might get 2 or 3 days over temperatures over 80 in a year. There's definitely a reason why your clients have holiday homes in California - it's most likely so they can escape the crappy UK 'summer'.
Sent from my iPhone using Tapatalk