Hi
There are 5 London boroughs who have put their cemetery records on Deceased Online (see the London burial guide for further details). They are Brent, Camden, Havering, Islington and Merton. In all there are 32 London boroughs plus the City of London. Three other London boroughs have their own cemetery records online Kingston-upon-Thames, Richmond-upon-Thames and Barking and Dagenham (though the latter is currently unavailable).
W7 would be the Paddington area therefore the City of Westminster (though Paddington Cemetery is now owned by Brent - records not online though).
http://en.wikipedia.org/wiki/W_postcode_areaIn SE London the two London boroughs for St Thomas' hospital would be Lambeth and Southwark. Southwark has three cemeteries as does Lambeth, plus an independently owned cemetery in Streatham.
Unless you are certain there was a burial you should also request a search through cremation records which are kept separately from burial registers.
If you are not sure of the cemetery, the search (excluding privately owned cemeteries) would be by London borough not by individual cemetery.
If he is buried with another family member that might widen the search otherwise those are the three most likely London boroughs to make a start with.
Regards
Valda