As you mention 1930-1980s, I think the system was broadly similar to today, for England & Wales. You mention the UK - rules are different for Scotland & Northern Ireland for inheritance where there is intestacy, I don't know about probate.
Broadly for England or Wales when a person dies if they leave a valid will the executor(s) go to a probate office and swear an oath to get authority (probate) to deal with the estate. The will then becomes a public document which can be accessed via the court service as on previous posts. If there isn't a valid will next or near kin go to the probate office for a
grand grant of letters of administration which then authorises them to deal with the estate under the rules of intestacy (these can be found on the court service website). All you get from the court service in such a case is the value of the estate and details of the administrators.
The probate calendars record for each year the grants of probate and letters of administration made in that calendar year (may be for deaths in the same or earlier years). I understand Ancestry have calendars upto around 1940 - after that local probate offices have the most recent ones and older ones may be found at record offices (not useful for you I realise). I believe there's a RootsChatter who goes into the main office in London & can search the calendars there.
For "small" estates a formal grant may not be needed (though some financial institutions may want one before releasing money).
For current rules here's a link (which also links through elsewhere for Scotland)
http://www.direct.gov.uk/en/Governmentcitizensandrights/Death/Preparation/DG_10029716(modified for spelling)
