Both are acceptable spellings; ιατρός is the one inherited from Ancient Greek and γιατρός is a more modern one. (So it's a little like colour/color or realize/realise, both of which are good and correct English spellings.)
In listening exercises, only one spelling can be accepted, unfortunately.
In sunny Cyprus the equivalent of the British NHS is the recently introduced GESY. It has had a few hiccups for the first few months but it's getting sorted. However it's not free. We pay 3 euros to see a GP and 6 euros for a specialist. Prescriptions are about 1 euro per item.
Waiting times to see a doctor are not instant. You might have to wait 29 minutes or more.
Applications to live in Cyprus need to be in soon. :-)