Hello, I am a solicitor with 20 years experience. I will try to help you with this.
The landlord wishing to sell his property is not a ground for possession. Your mother is under no obligation to accept any offer and does not have to leave for this reason. In terms of price this is a matter of negotiation if your mother agrees to leave. The usual starting point for negotiations is the difference in value between the property with a protected tenant and the property without. This may be (but might not be) more than £20,000. In terms of time this is up for negotiation, there is no time limit imposed by law.
However you should be aware that in some circumstances the Landlord can obtain possession on the basis that he, or a member of his family or if the original landlord is dead his heir wishes to return and live in the property. If this is a risk then it may be possible for the landlord if negotiations with your mother fail that the landlord etc. may claim he wishes to move in to the property. This will not be straightforward for the landlord but may be a risk.