Hello MissBeeby
See post above a few hours ago.
Some Wills and Administrations are in "The Gazette" online ...
(The Gazette - the UK's official public record since 1665.)
Including some who died Intestate (those who left no Will) and other Public Notices relating to people.
No Log in required and it is free to search.
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The person who was 4 years old at time, do they not have any other relatives, who might have a piece of the information you are seeking?
Paying out £26 with one Cemetery authority, if they were buried or cremated in another authority area, a family grave elsewhere, where they were born, or a denominational cemetery, might be a waste of money.
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Some Funeral Directors or Undertakers keep records and sometimes some are helpful and as you have a name AND exact date of death, you could try writing to those in the area, also stating your relationship to the deceased etc., and you may get a reply.
I traced the family of a WW 2 Airman (who flew with my Grandfather, killed in 1940) by writing very polite letters to all the Funeral Homes in a Canadian town where he came from, and enclosing some photocopy proof our family had, that my Grandfather was in the RAF and with the same Squadron and RAF Station.
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You may need a lot of patience to trace these burials or cremations.
Mark