Good morning,
There is NO national health service in Cyprus.
You don't say of you are receiving state pension or not.
If you are not- Obtain a S1 (formally E106) from your tax office prior to leaving UK. Depending on you NI contributions this can give you up to 2.5 years free healthcare in Cyprus.
If you are prior to leaving the UK obtain an S1 (formally E121) from:
International Pension Centre
Tyneview Park
Newcastle Upon Tyne
NE98 1BA
United Kingdom
You will be entitled to free healthcare (although not all medications are available free) from a state hospital or state doctor. Some medications are not available from the state, and need to be either purchased over the counter at a pharmacy or by a private prescription (all psychiatric drugs and schedule 2 & 3 drugs need a private prescription)
If you have a partner who is below retirement age they may be entitled to free healthcare under your card.
Make sure you have a copy of all your medical records from your UK GP. (Once you leave UK and are resident in Cyprus your GP medical records are centrally stored in UK for only 10 years)
Ask your UK general practitioner for a supply of your usual medications prior to moving over and ensure that you have the prescription to back up the medications you are on.
Until you get your relevant healthcare card you may need to seek a repeat prescription.
Kind regards