Just a slight nudge to the timeframe: Kelly's 1937 Directory of Beckenham & Penge has Mrs Thompson at 6 Maple Rd, as in the 1939 register. For some reason, the house isn't shown in the 1938 or 1939 directories, but she is in the 1939 electoral roll as Lilian Thompson (no Joseph, but other unrelated people, as in the 1939 register). Going backwards, the resident at No 6 is George Shepherdson from 1924 to 1936, and Joshua Bedford in 1923. Because the house was in multiple occupation and directories generally only show one resident, this doesn't prove she wasn't there also. To check further would need earlier ERs, which I don't have, but you could ask
Bromley Local Studies to look for you. Tell them it's in Ward FF, as this long road was split between two. They may not charge for a quick lookup.
Thinking the Thompsons might have moved within the area covered by the directories, I checked on Mrs Thompsons (first names weren't usually given for women) and Joseph Thompsons in a few years before 1937, but eliminated all possibilities (e.g. they were also shown in 1937).
My guess after all that would be that she moved to Penge for some reason shortly after Joseph's death, which could have been around 1936, and this may have been elsewhere. If he did die in Penge, his most likely resting place would be Beckenham Crematorium & Cemetery. No Penge churches have burials.
