Use it. When I found a lump last year I asked the GP to make the private referral, and she said "trust me, this is where the NHS gets it right, you don't need to go private".
I had waited a week to see the GP, then had the full standard two week wait for an appt (at a hospital nearly an hour's drive away as all the others were overbooked), but then the wait for biopsy results took far longer than it was supposed to, appointments were cancelled at short notice due to a lack of staff, in the end I had to beg them to tell me over the phone. The anxiety over that 6 week period was unbearable.
Fortunately all was fine, it was benign. But if I'd gone private my local clinic would have seen me in under a week and likely had the biopsy results within a week or so too, far reducing the wait and the stress.
I received excellent care, the staff were wonderful at every stage, I don't think private would have been 'better', but tracking down the relevant dept to chase stuff when things were cancelled was hard. I also felt like if I'd gone private I would have freed up a spot in the queue for someone else.
Either way, hope you get seen soon.