When crossing the border between the north and south of the island, stamps aren't put in the passport. For the southern part of Cyprus, this border is considered internal. The North agrees with this system. The data of the entrants are checked only against the database.
But in order to get from the UK to the Republic of Cyprus, you need an EU passport, a residence permit / permanent residence of one of the EU countries or a multiple entry visa for Bulgaria, Romania, Croatia, the Republic of Cyprus.
