If you can afford boarding, he will find a place at senior school. Also many boarding schools that are near to London empty out most weekends even if they're not officially weekly boarding so I wouldn't discount full boarding initially.
For a very long list of potential schools, you could do worse than look at the leavers destinations for Bruern Abbey prep, which specialises in boys with specific learning difficulties like dyslexia (not just the senior school itself). Even though you're working with a different diagnosis, I expect parents of boys at Bruern are looking for many of the same things such as good pastoral care. It's a broad range of schools, but you can probably discount the ones you already know won't be right or are too far away to make it more manageable https://www.bruernabbey.org/about-bruern-abbey-school/leavers-destinations
Milton Abbey, Clayesmore, Bloxham and Shiplake are on that list and all offer BTEC as well as A-level which, if your son continues to struggle with exams, might suit him really well in the longer term as they are less exam-focused qualifications. I'm sure there are other schools listed there that do, as well as the more academic options.
Also, don't discount state schools. They are often better resourced and skilled at working with pupils with ADHD than private schools and in London there are many good state schools. If you are looking for a place in London outside the main Y7 entry then you'll find something. London is transient and schools have to accept you if they have a vacant place.
It's difficult to know from what you say whether your son is above or below average academically compared to a broader population. It is certainly possible to be average or below at Wetherby, but be able to get above average results in the right senior school. Every child deserves a school where their particular strengths are valued and they can achieve their potential.