To add a bit more specificity to OP's question, the mascot for the tournament, World Cup Willie, wears a Union Jack even though the 1966 World Cup was hosted exclusively in England. So not only how long did fans choose to raise the Union Jack, but how long was it designated as the formal representation of England?

This is a hard question to answer as the change didn't happen at once. In fact at England games today, you'll still see the Union Flag flying. You can see here in the Panini album from 1980 they use the Union Flag on the cover and the St George Cross for the players. During the times of the changeover, the England team wasn't very successful, so it is quite hard to find sources. After the world cup win in 1966, England went to Mexico as holders in 1970, and then missed out on the 1974 and 1978 tournaments, returning in 1982. In the European championships, they missed out on the 1972 and 1976 tournaments too. The FA uses three lions as its symbol and not a flag, so it's not easy to tell there.

If I was to guess based on my own experience I'd say it was a gradual change from the early 1970s, based on the growth of nationalism in other UK countries. Each year till 1984 (and till 1989 in the Rous cup) the four home countries of the UK played in the Home Internationals tournament. The Scots and the Welsh booed God Save the Queen as an anthem before the games, and gradually adopted their own anthems. At the same time, the English support, aware of their own unique nationalism, moved from the Union Flag to the George Cross,