The Liberal strength in Bethnal Green was down to Percy Harris, Liberal MP for Bethnal Green SW from 1922-45.
Harris's victory in 1935 was, I think, the last time the Liberals won a Westminster seat in the LCC area until the 1983 Bermondsey by-election. After his Westminster defeat in 1945 he was returned for the LCC in 1946, holding his seat until his death and the balance of power after the 1949 elections. He had a final shot at the (now merged) Bethnal Green seat in 1950 but came second again. Roy Douglas took over the candidacy in 1951 and held second place but slipped to third in 1955. Despite staying in third until the mid 1970s the Liberals contested the seat at every subsequent general election, something that didn't happen in many places, even those where they'd had a post war victory.
Boundary changes to Bethnal Green, which added the Hackney wards of Victoria, Triangle and Wick to it (from 1950-55 it was just Bethnal Green MB), sapped the constituency's Liberal strength further.