[Apologies to Tedscout - I was typing as your post was submitted]

Hello Tricia,
From the info you have given, we know that your grandmother was born circa 1926. There is only one birth in Englad and Wales for that year in the name of
Patricia J Underwood, and that was registered at Kensington (Greater London) in the June quarter, vol. 1a, page 177.
The mother's maiden name is given as
Hill. A marriage search reveals only one marriage for an
Alfred Thomas Underwood, and that is to an
Amelia Caroline Hill, at St. Mark's Church, Poplar, on the 3rd August 1902.
Alfred was aged 24 (Blacksmith) and
Amelia was aged 20.
Parents' names given as:
John Henry Underwood (Blacksmith)
Leopold Hill (Shoemaker).
I can find only four deaths in the name of
Patricia J Smith post-1953, only one of which is in the London area. Here are all four:
Patricia J Smith, 1952, Romsey, Hampshire (too early)
Patricia J Smith, 1960, Plymouth, Devonshire
Patricia J Smith, 1965, Tonbridge, Kent
Patricia J Smith, 1962, London, City Greater London Hope this is of some use.
Regards,
Peter.
[Edit: In addition to Tedscout's post, there is also the death of a
Patricia Joan Smith (b. circa 1926) Mar 1993 Chelmsford Essex - but is of course too late].