I think your only option, if you are 100% certain this is what you want, is to do an MBA at a top 20 (global!) business school and go from there. It would be £70k minimum fees alone, plus not earning for two years, and the chance that you don’t actually graduate or find a job afterwards.
At your age I’m afraid your lack of experience is really going to count against you - a competitive bank can hire straight out of uni, a PE can hire a 2nd year bank analyst who is younger, more happy to work til 2am and on weekends, and can be moulded more easily.
A CFA 1 will show some dedication but frankly most candidates will have been Cfa chartered by their 3rd year into this career path.
If it’s not the investing/front office side you want, but just to work at a PE/bank, have a think about moving to compliance/middle office functions. I imagine (outside unforeseen problems!) the hours are more regular and shorter than accountancy, and the pay is probably decent relatively (can go into 6 figures with experience, plus a decent bonus on top).
With your experience and a CFa1, if you are VERY, VERY, VERY lucky, and cast your nets wide and ace all the interviews/tests, you might be recruited as an analyst in the front office at a smaller asset management company. Do well there for 5 years and you might have more options to move to PE/bank, but frankly mostly mid 30s are looking to leave PE/bank into buy side.
Sorry if this doesn’t sound like a bed of roses, I’ve been brutally honest here, and that’s the first thing you have to be able to withstand at a PE/bank. On a personal level though, whatever you choose I wish you lots of luck.