Unlikely. Your grandfather and/or mother would have lost any claim to British nationality after the
Cyprus Act 1960, if their only connection to British nationality was via Cyprus (i.e. they did not have a claim to British nationality via another colony or via the UK itself).
They would also have needed a connection with the UK itself (via either birth, registration, naturalisation or five years residence before 1983) and not one of the colonies, to qualify for Right of Abode after the Immigration Act 1971. If they did not have such a connection to the UK, they would not have become British citizens in 1983. Most categories of British nationality other than British citizenship can't be transmitted to future generations at all.
Did either your grandfather or mother have such a connection with the UK before 1983?
PS: As an aside, the Sovereign Base Areas of Akrotiri and Dhekelia do not count as British territory for the transmission of British citizenship/nationality even now, as they are military bases and not Overseas Territories (and thus their legal status is different).
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.