OP, you and your mother need to put your big girl pants on and start dealing with this, rather than burying your heads in the sand and trying to explain your way out of this with the “old age” and “am unemployed” excuses. If your mother cannot deal with this, you will have to on her behalf, and being unemployed will surely mean you will have plenty of time to do so.
When a leasehold property is being sold, the buyer’s solicitor has to legally ask for responses to leasehold enquiries. This is usually done with a standardised form called the LPE1. This form is meant to cover all forms of leasehold property and is mostly applicable to flats/apartments, rather than leasehold houses.
As you are being asked to answer questions regarding the sale of a leasehold house (not a flat) for which you are the freeholder, a lot of the questions will not apply to you.
It would, however, be in your best interests if you could look at the form question by question and answer as best you can the questions which do apply to you. If many of the questions are not applicable, then state Not Applicable.
You would be well advised to obtain a copy of the Lease and the title documents (both leasehold and freehold) for the neighbour’s property to see what it says about who is responsible for what. It will also state the start date of the Lease and how long the term is. It might have been for 999 years, making the property almost freehold. If this is the case, it might be advisable to offer to sell the freehold at an appropriate price. You would need to instruct a solicitor experienced in leasehold to carry out the process for you.
The buyer’s solicitor will not visit the property, so they have to rely on the answers you give. They don’t know if you collect ground rent, service charges, insure the property or whether the buyer has to enter into a Deed of Covenant to agree to any covenants in the lease. Therefore, as the freeholder, you should answer them.
While it isn’t compulsory to complete the form, the sale is likely to fall through if you don’t, as lenders won’t lend unless the questions are answered. Lenders won’t want to be financially liable if a previous leaseholder owed money for not paying his service charges. While that is obviously not the case here, the lender doesn’t know that. Lack of answers will slow the conveyancing down considerably.
If you want to be obstructive/unhelpful because the neighbour has been obstructive/unhelpful to you, then the sale will go nowhere and you’ll be stuck with these neighbours for a long time. Surely, you’d be pleased to see the back of them?
You have consented to the work they’ve done on the property, because you didn’t object when they started the work. That was the time to object and to force them into a Party Wall Agreement.
You have had a problem with water ingress and damaged steps, so maybe use this to get them to pay up for the work, before you agree to complete the form. Paying you money to sort it yourself will be a lot quicker than expecting them to actually do the work at this time of year and to a decent standard.
I suggest you take on board all the (free) advice that posters have given you in their own time and start taking responsibility for being the freeholder that you are.
As for not wanting to give your name or contact details to the buyer's solicitor, due to confidentiality issues, you are being ridiculous. As a freeholder of the neighbour's property, you are a party to the transaction and, as such, they have every right to contact you for information pertaining to that property. I would expect your details are stated on the Freehold Title Register held at Land Registry anyway.