In the times of Cold War, Cyprus was an important base for NATO and UK therefore to keep them there with none-to-low amount of legitimacy it was beneficial to keep Cyprus somehow divided.

There are also 'conspiracy theories' that British intelligence had agenda of creating unrest intentionally on the island so they can keep the bases, but that sounds too much of 4d chess, since this unrest led into Greece leaving military part of NATO and made western alliance look shaky against USSR.

also worth reading : http://news.bbc.co.uk/2/hi/europe/4632080.stm