OP I have been there in london.
TBH I have ended up saving up and using Pimlico plumbers. At least they do come and then you have a way of getting in touch with them if something goes to shit in the following year (it has for me before and Pimlico remedied the issue). Tbh the joy of phoning and someone answering that phone is something else in plumber land!!
Then phrase the job as ‘replacement of all rads, check pipework upgrade as necessary, etc etc) so that you sound like you appreciate it could be a difficult job more complicated than it first seems. I think sometimes trades get put off by customers pitching it as a straightforward job as they’re already thinking ‘i bet I get there, take out 2 ancient rads and then there’ll be all sorts of other work and she’ll be like ‘that’s not what I asked for’ etc’
I would buy a couple of efficient little plug in oil rads to get you to spring and then save up so you’re doing all the radiators you ever want done in one go, plus get your system flushed (it gets the sludge out basically), any pipe work etc.
I also really recommend anyone in london to do a course at the good life centre in diy and plumbing. I did one on general diy (we have moved out of london now but I am on my 6th owned house 4 of them reno jobs)
I MORE than made the money back in a year based on not needing to call out people for lots of different odds and end jobs. Subsequent builders have actually commented on how good my work is!
It means you save money on the small jobs and so have that leeway to throw a bit more at harder jobs.
Its so difficult and frustrating I know! Absolutely none of my advice is perfect but is how I got stuff done in the end.
Also advice from neighbours as if you have anyone reasonable local and they don’t have to commute for your job, you’re more attractive. State that there is parking for their van etc. I always chat to tradespeople, I find all the trades fascinating, and over the years have picked up so many things that I wouldn’t have thought of (that parking is a big issue is a fairly obvious one!)
good luck!