You say it's cheeky, but the way I see it, anything you can get done privately will take some pressure off the NHS, so they shouldn't object. Also, a lot of surgeons work for both the NHS and private, and will have contacts for both, it would be great to get one of those! I saw my surgeon privately, but if I'd joined the waiting list I could have had the same thing done on the NHS, just in a different hospital (literally across the road!).
I'd definitely go to your GP as soon as you can to try and coordinate this. The fact you're being made redundant makes it clear you're not just after a queue skip, a good GP should be helpful in looking at a long term plan. Find out about after-surgery care; if it is just physiotherapy then how long is the waiting list, and can you join it now. Also, how many times do you need to see the surgeon after surgery? This is the really expensive part of treatment, and the thing you need to able to get on the NHS immediately since it's so important. I guess you may need to get on the waiting list for after-surgery care now, to make sure there is no gap in your treatment. I'm not sure.
Obviously your financial situation may be temporarily uncertain, so please don't consider this insensitive if it's completely inappropriate for you, I say this only with your knees' interests at heart! ...and I'm an accountant so increasing costs doesn't come naturally, I genuinely think it may be worth it. If you are looking at a gap between insurance ending and NHS care starting, you may find that self-funding private care for a month or two is not as expensive as you think. My London priced physio sessions were £40 each, one or two a week. Also, take into account that being on crutches may reduce your outgoings; frankly by being confined to the house I probably saved about £20 a week I'd normally spend on dinner or at the pub anyway, plus £30 train tickets! (I was able to work from home for six weeks). If the gap in treatment when you are not yet at the top of the NHS waiting list is only a month, you may only spend £400, and if it gets you back to work a month quicker it may be worth just paying. It's the £200 surgeon consultations that will rake you over the coals financially, hence the importance of getting a surgeon who will see you promptly and on the NHS for later consultations.
Note also I've never taken my treatment from private to NHS, I've been going the other way, but as a taxpayer I sincerely hope the NHS has the foresight to get you the best treatment it can through the private sector first, and then support you afterwards.
Good luck and all my best wishes for our beautiful, clunky health system and your no doubt beautiful, clunky knees!
Lulu