Swords aren't considered knives, they are considered (correctly) to be bladed, and thus to fall under the scope of various bits of legislation, like this: https://www.legislation.gov.uk/ukpga/2019/17/section/41
You're right that they say "knife or weapon". So they don't classify a knife as a sword
There is also no London specific governing body
Swords aren't considered knives, they are considered (correctly) to be bladed, and thus to fall under the scope of various bits of legislation, like this: https://www.legislation.gov.uk/ukpga/2019/17/section/41