OP what area of London are you in? I have a private slt for one of mine who is very good, I got her based on other parent's recommendation and she focuses more on a specific type of speech disorder but if your dd's problem is in speech sound production rather than simply speech delay then that is more what this slt does. Paying 70 per hour (more for assessment) and did have a bit of a waiting list but not too bad.
I got a different one before just by going off the helpwithtalking website and as you don't have any recommendations from other parents you don't really know if they are any good or not? That one was really disappointing, worse than any I have had on nhs and just wasn't happy with that experience.
Others from helpwithtalking site did seem good as I phoned and spoke to them about dses needs and they were happy to talk quite at length on the phone for free or give a bit of advice about who/what else they could recommend.
I have found that majority of the good ones will have a waiting list even on private, so if they are tooooo available perhaps its not always a good sign?
Have you thought of trying to apply for a statement for her? If she is starting school in sept and incomprehensible then she needs one, because she will need a significant amount of extra support, and there are dedicated speech units in some schools too if she needed that level but i think she would need a statement to access it?
Theres also the nuffield which is on nhs and focuses on speech sound production difficulties, apparently its really good. Not been able to get my ds there (yet! im still bearing it in mind for future!!) cos he also has asd but it could be worth a try for your daughter. You can go and ask the gp for referral to them apparently or phone them directly and ask advice, i think the name is pam williams? someone correct me if im wrong?
Also worth phoning your local slt dept again and just saying that she has been 10 months on the initial waiting list, she starts school in sept, and has now been put to wait again and she needs to be seen urgently as you're really concerned - be polite but firm sometimes it works wonders to just make yourself a bit of a pain in the ass (while remaining polite so as not to piss them off!!) those who speak up a lot do get better provision than those who don't. that's quite sad but its very much the case. obviously there are still waiting lists but a bit of a nudge sometimes seems to make them suddenly realise that oh yes actually there is a place for her in a group or whatever else!
Are you on facebook theres a group for verbal dyspraxia support, lot of mn people are on it and the collective knowledge and experience of the parents on there is pretty helpful! this is where I got my private slt recommendation from, I would say its worth joining.