I am also quite baffled by your original post or why you have both got into such a tailspin about this.
First question, has the probate even gone through yet? If not, it's not your friend's house yet. If not, he can't really do much about anything.
Second, what your friend decides to do about doing up/selling the property is none of the business of the tenant over the road, his impending homelessness is not your friend's problem. "The house is not on the market at this time, you will need to make other arrangements". End of story.
If this guy is going to be a problem, I would suggest installing a camera and extra locks as soon as the probate goes through to prevent him going in and squatting.
Thirdly, why on earth would doing something about an overhanging tree cost £20,000? The neighbour can cut back anything overhanging themselves and put the wood/cuttings over the fence. If this is about a tree surgeon, it's not going to cost £20,000. If it's actually about tree preservation orders or roots undermining foundations, that's a different story.
Fourthly, once he fully takes ownership, he needs to get a valuation on (a) the property now without any work being done, (b) the property done up to basic liveable standard, (c) quote for the work to get it from (a) to (b).
This will inform him about the best course of action in terms of what to do about works and sale. It might be that he is better to sell it as a wreck at auction (like the homes on Homes under the Hammer, where trasdemen buy them and do the work themselves at their own cost) and just get rid of it and get what he can for it in the state that it is in. However, it may be that if he gets the work done he can get the money back with extra from the sale, so it may be worthwhile to do that, and there is no real time pressure to do so.
Fifthly, regardless of the option taken under (4) above, if his mother was a hoarder it seems likely that he will need a house clearance company. He needs to go through the contents and extract anything of value that might be able to sell to help cover costs. Once have dealt with any literal rubbish, there are charities that do this for a fixed amount that they will price - they literally come with a van and take the lot. They will then sell what they can and deal with the disposal of the rest.