That's not a great comparison although yes, Queenstown is manic if you go when everyone else does (pro tip: do not do this, go in off season). The population density of NZ is 20 per Km2 (51 people per mi2). The population density of the UK is 280 per Km2 (725 people per mi2).
To keep things vaguely on topic the population density in Cyprus is 136 per Km2 (353 people per mi2), so about half of the UK.
It's also a tiny island with people generally living on the coast or in Nicosia. Otherwise you could have many kilometres between you and your nearest neighbour. Small towns are small. There's more expansion in large cities than tiny towns.
The tourist traps are just that and people move there. Ayia Napa is a dump now (it used to be a hotel, a handful of villas and a little high street). Now it feels like chav central and other than the weather you feel you're in the UK.
Limassol is a very industrial town and almost the second capital, Paphos (pronounced Pah-fos, not Pah-thos, sorry bugbear of mine
) is again now just a tourist area, Nicosia is the capital and also where a lot of the work gets done but of course more crowded and chaotic. Larnaca is a blend of tourism and not but because it has a beach is still tourist heavy.
The island is so small you can get anywhere within an hour or two. I mean Nicosia to Larnaca is 30 mins with no traffic. I have friends who commute to Limassol from Nicosia takes him less than an hour.
Life starts early as it's cooler and often things shut down for lunch and reopen in afternoon till later. However it's becoming more western with generally longer opening hours now.
As a Medical Doctor you'd have a lot of opportunity there but I don't know enough about the industry to know if you'll need to do any courses etc to convert or demonstrate capability. It's an EU country so will follow EU guidance (which is helpful as.it means you can move around the EU easily to do the same thing).
GPs are always busy there but far more efficient and effective than the UK. Healthcare in Cyprus is great. If ever I have a niggle whilst I'm out there I always go and they sort me right out. You get x-ray results same day and you actually keep a copy of it too. You can pretty much on a whim see any specialist from an ENT to a midwife to an Osteopath same day pretty much. Medical doctors can earn well over €400k in Cyprus depending on experience and seniority.