North Cyprus has two mobile service operators, namely KKTCell (Kuzey Kıbrıs Turkcell) which everyone calls Turkcell which is actually the parent company based in Turkey. Then there’s Telsim which is now owned in part by Vodafone. Mobile numbers generaly start with 0553, 0542, 0548 or 0539.
Both companies operate on GSM, 850, 900, 1800, and 1900 which most modern day mobile phones seem to be configured for (unless you still own a brick that is!), so you should have no problems in North Cyprus. If you want to use your UK number whilst you are holidaying here, then KKTCell and Telsim both have agreements in place with all major worldwide networks, so you should not have a problem connecting to their networks on your arrival.
Roaming fees however can be very high at times, so if you are planning on staying in North Cyprus for long periods, then you are much better off buying a pay as you go sim
KKTCells and Telsims’ websites are KKTCell (which does have an English version) and KKTC Telsim (only in Turkish currently).