There are quite a few options around London which the OP hasn't mentioned and as someone who's also thinking of doing a PPL in the SE, there's a few below I can think of. You don't necessrily need to do a PPL (and maybe modular too) or hours building at the flying school where you intend to do a CPL/ME/IR.
Also, how much have you flown so far? Do you know that you enjoy it/have the aptitude? Do you have the spare time to commit to ATPL theory? I'm looking at doing the PPL standalone first, effectively treating it as an enjoyability/aptitude/spare time test and then taking things from there if it works out.
Bear in mind some of the flying schools which teach the commercial syllabus will prioritise commercial students when it comes to instructor/aircraft availability, so as a PPL student this is something you may want to think about.
NWFT at North Weald seem to be very competitive on price and come across as more of a club than a school, so to speak. Just off the M25 or a taxi from Epping station on the Central Line.
I'd imagine Redhill wouldn't be too difficult for the OP to reach (Tube or Southern to London Bridge, then Thameslink for circa. 45 minutes.
Elstree probably has a similar journey time to Redhill but of course in the opposite direction.
White Waltham could be doable by Jubilee Line from Bermondsey to Baker Street, then Bakerloo to Paddington and a GWR fast service to Maidenhead (half an hour). Likewise, Booker from Marylebone is an option.
If you live outside of London, driving to the airfield is usually a no-brainer, however getting the train from London allows you time to relax and to think about the upcoming training whilst planning things out. The return journey can be a good time to mull things over, I definitely wouldn't rule out getting the train to the airfield and that opens up the choices a bit (I should add that I'm a bit biased because I get a hefty discount off walk-up rail travel). It also allows you to avoid the crazy drivers in/around London and their suicidal overtaking manoeuvres. Some of the airfields will require a taxi to reach from the station, some won't.
It also depends on how the OP plans their training, if you intend to stay at the airfield all day. For example, turn up, fly a trip of one hour or more, take a break then fly a second trip (and even possibly a third), then distance to the airfield is of course less of a problem than if you just intend to do an hour and go home.
Lastly, one thing that makes me scratch my head is the pricing disparity between flying clubs/schools, how can some charge £150/hour dual in a C152 and how come others charge £60 more? I happened to have a trial lesson at one of the latter schools and the aircraft was still fairly tired if you ask me.