To be pedantic, both 0171 and 0181 became 020, but with the 7 or the 8 moved to the front of the number, so 0171 222 1234 [1] became 020 7222 1234, with 7222 1234 being dialable in London without the area code.

There are also now London numbers that start with a 3 or a 4 as well as 7 and 8 so it's important to properly describe the dialling code for London as 020.

Misconceptions about telephone dialling in the UK are so commonplace that they merit their own Wikipedia page: https://en.wikipedia.org/wiki/UK_telephone_code_misconceptio...

[1] a real number that will get you through to Transport for London enquiries