@Bongabob
Ay up, lad...
Congrats on finding a nice house in Bulgaria, and good luck with the move! Hope you enjoy Bulgaria as much as I do.
There's also a couple of quite interesting houses on Ebay at the moment, if you're still keeping an open mind.
Sorry, I'm not 100% sure what the process is for Brits here after Brexit, when they have EU residence elsewhere. With UK passport only it's the dreaded D visa. But if you have a Hungarian permanent residence permit (after 5 years), it should be very easy, and direct to residence. If you have the temporary/initial one, it might not be a sure thing. I think I'd be tempted to scan my Hungarian card and have a quick consult with an immigration lawyer (I can recommend one, if you want).
For finances, I haven't checked the latest requirements, but I know the statutory minimum wage is now 360 euros per month. So certainly they can't ask for more than a year of that (about 4,300 euros)! And probably less will be fine. For me they accepted a notarized declaration (summary: "I have money, I won't ask Bulgaria for anything") and a copy of my debit card. :-) So 50k.... PLUS an income (which I'd guess is more than the minimum wage)... I can't imagine they'd have any issues.
Banks here used to be a bit wild, but the EU put paid to that. You might be able to shop around and find one that will open an account just with your UK passport, but most now want to see your BG residence card, so it will be much easier after you've got your official residence sorted out. They also ask, in general, where your money comes from. And, in particular, for larger transfers. Basically, a few grand shouldn't raise any eyebrows... and as you've sold your house you'll have some official notary documents for 87k, or close to that... so you'd be able to say "house sale" and transfer some of that here without any hassles.
Aside from proof of finances, they want proof of health insurance (EHIC from Hungary or UK, or a local policy for 100 euros) and proof of address (the notary deed for your house after you buy it is the usual one).
Property costs are relatively low here, and the cost of living is pretty low too. If you can buy a house for cash, and still have 50k leftover... plus a UK income... you should be able to live here very comfortably. Especially if you've also got a private/state pension to look forward to.