I think it depends.
A couple of months isn’t very long in terms of building a therapeutic alliance and opening up and digging around in complex, traumatic stuff. But if you have a particular issue or symptom etc that you’re specifically struggling with at the moment, it is certainly possible that you could make progress with that and get some useful support while you wait for your NHS therapy to start.
For me, I think if I could narrow a focus to a particular challenge to work on to make day to day life more manageable (eg anxiety) that would mean I would be less concerned about the need/want for a strong therapeutic relationship and trust would be less of an issue.
Personally, I do feel it is important that I trust my therapist. And my medical professionals. And I find that very difficult sometimes. But that’s probably because doctors and teachers have been abusive in my past - hence part of the reason why I am in therapy in the first place!
Your NHS therapy is likely to be short term, I suspect. So, I guess another plus of starting with a private one now means you may identify someone who you would like to return to and continue with after your NHS sessions have run out.
Ultimately though, I think it depends on your financial position. If you really can’t afford private therapy, spending money that you don’t have on it probably isn’t going to make anything better. If anything, worrying about money or getting into debt will probably add to your struggles.
So, perhaps start with that - can you realistically afford some private sessions or not? And, if not, what other resources can you use for support to bridge the gap between now and your NHS therapy starting? This place? Reading? YouTube videos? Are there any low cost groups in your area?
If you think you can afford it in the short term, perhaps think about what particular thing you would benefit from focusing on. Where would you like to get to with that issue in your time with the private therapist?
And, on the trust front, it may be worth asking yourself what you might need in order to feel safer/trust in therapy. And, if you can identify something, then look to see how you might go about meeting that need. It may be something you can do for yourself. It might be something you can ask a therapist about or something that can help you choose a private therapist if you go down the private route.