To clarify the road tax issue. All bikes in the UK need to have it before they can be ridden on our roads. However, chances are, the bike you buy should already have some on it if is being used on the roads at the moment (it is renewed either every 6mths or 12mths). If not, the current owner could probably tax it for 6months (£36.30) in their name if they still have a current insurance policy on the bike.
If the bike is less than 3yrs old it will not need an MOT certificate, older than that and it will need one but to be honest, if I was buying a bike over 3yrs old I would expect it to have the MOT certificate - be wary if it doesn't!
Regarding insurance - I doubt you would find any insurance company in the UK who would insure you if you do not have permanent address here so the easiest option would be to insure it in Cyprus then add European travel to the policy if it isn't already included.
Of course, if you decide to buy new it's a different set of rules to do with VATand road tax but the dealer you buy from can arrange 'exempt' (ie free) road tax which allows the bike to be ridden on the roads here providing it is exported within a certain time limit (2mths I believe). This does require a certain amount of form filling (VAT411 if I remember rightly) but on the plus side you can buy it without paying VAT in the UK, although you have to pay it on importation to Cyprus or other EU country. Should you for some reason decide to stay in our wonderful country and the 2mths pass without the vehicle being exported, you will receive a hefty bill for the VAT and the registration will be cancelled. Same goes if the bike should be written off before you get it out of the country - you'd still get a bill for the VAT - so make sure your insurance covers the full value of the bike including the tax should you buy a new one.
Good luck!