Hello I am a solicitor with 20 years experience. I will try to help you with this.
As far as the fence is concerned, if it is over hanging your land it should not do so, it should certainly not touch your father's house.
Any branches or leaves which overhang your land can be cut down, although you must offer the cuttings back to the neighbour.
Assuming the fence is the neighbour's legal responsibility you can ask him to fix it but if he doesn't and in order to avoid any dispute you might need to declare in a sale you could simply get it fixed from your side if that is at all practical. This would be at your expense which would no doubt be irritating.
Unfortunately, as far as the right to light is concerned, the 20 year rule you refer to regarding the fence will also apply to rights to light blocked by trees.
So can our neighbour refuse us access to his property to undergo maintenance ad repairs?
Has that work been carried out? I thought that had been done form your question.
In brief the legal position about that is as follows.
There may be a right of entry specifically for the purposes of inspection or repair in your father's property’s legal documents. If there is no such right, the law allows you as the person wishing to carry out repairs to apply to the county court for an access order allowing you to enter your neighbour’s land to carry out the repairs. There is a fee for the application. If successful your neighbour would have to pay the costs. If there is a right in the legal documents for access to his land and he has unreasonably blocked access it may be possible to gain compensation for any resulting harm.