Get yourself sorted for 3 months. I don't think any mortgage adviser will look further back than this.
They are looking for financial commitments, credit cards, loans etc. They don't care or need to know before this. You could say your a wargaming geek, but tbh they won't even ask (assuming you don't have massive debts and then it wouldn't matter what you say).
£20k isn't crap as a deposit. It would get you a lower end property, but at least it's sinking into capital. For best deals with apr %, you're looking for the £120k ish but could go up to £200k plus, if your annual income is high enough.
This seeking arrangement things sounds dubious. It sounds like you're living with your parents, but trying to act like your James Bond? Knock that on the head, at least with escorts it's a transactional basis not ongoing costs.
You probably don't even need a mortgage adviser tbh. GO on Experian and get a credit score, to see if you are likely to get credit. Then go on a comparasion site or specifically YBS or Nationwide. See somewhere you like, then ask them to drop at least £5k. You're be a 1st time buyer, so no chain. Drive around property at late evening time to see what the neighbourhood are like.
If you want a flat, ask for how long the freehold is (preferably 100 years) and check the management fees per month for the guys who run the freehold.