It is very likely that the design of the People’s Action Party’s logo was based on the British Union of Fascists.

For context, the British Union of Fascists was formed in 1932. Lee Kuan Yew, the leader of the People’s Action Party, was studying in Cambridge University in the 1940s. The People’s Action Party was formed in November 1954 and would debut in the 1955 general election.

In Singapore Struggle for Success, John Gordon Stewart Drysdale, a British officer who served in Singapore and later lived there writes:

By early 1955, some weeks before the general election, which was scheduled for April 2, the People’s Action Party became the butt of jokes and criticism. Even the party’s symbol - a flash of lightning in a circle - was claimed to be a copy of the symbol used by the pre-war Fascist leader in Great Britain, Sir Oswald Mosley.

As this was during the early days of the People’s Action Party’s formation, and the knowledge that their symbol was similar to the British Union of Fascists was somewhat common knowledge, it would be very unlikely that the party leaders were unaware of the similarity.

There is a secondary source which claim Lee Kuan Yew deliberately adopted elements of the design, but I am unable to find the primary source.

A report in the financial times on an interview by the London Observer with Lee Kuan Yew states:

He acknowledged a design influence from the BUF symbol but added that the colour scheme had been changed for the PAP logo.

If the report by the financial times is accurate, then the leaders of the People’s Action Party did base their symbol’s design on the British Union of Fascists.