1939 registers are available on F M P and Ancestry, I think they may sometimes appear as part of their free weekends. FindMyPast updates theirs regularly, Ancestry once a year.
Also worth mentioning that access is free at many libraries, and some authorities have arranged for free access to Ancestry for library members (at home) due to the current Covid-related closure of libraries.
If I pay how easily can I search for those - do I first have to find out which registration district they were in or can I just search the whole database with surname and given names?
Searching is essentially the same as for the online census records - if the name(s) are unusual then you might be lucky and get away with those only.
There is no birthplace information, and the records don't give the name of the parish someone was living in, so it is necessary to have an understanding of the registration districts to work out whether a result might be correct.
The compensation is having the full date of birth available sometimes helps - although only if it is correct. From my own searching I've found there is a good chance of the year being +/-1 and it is surprising (to me at least) how many people's birth days don't match other sources.
Also bear in mind there was already a lot of war-related displacement of people. Either evacuees or forces families might not be where you expect them to be, and forces personnel are generally missing entirely.
And finally it is worth bearing in mind that you don't necessarily need to pay in order to just
search the records. E.g. F M P still operate on the principle of non-subscribers paying to 'unlock' individual households - but typically it is possible to locate a household of interest before payment is required. Therefore if you are only going to subscribe for a short period of time (or will use 'free' access opportunities) then it makes sense to use the free searching options to first locate the households you are interested in (and note down the details), so you can be more productive when you have paid for/free access to the full information.
There is still a lot of information online from when the register was first launched about how to make the most of the searching facilities and to improve the chances you've found the right household before paying to view. Much of this is still relevant if you don't have full subs.