It’s not in general illegal to carry knives in London (or again, in the rest of the country, which has the same laws). Small knives are permitted generally and larger knives may be carried for specific reasons (e.g. religious). To say that it’s illegal to carry a knife in the UK is roughly as misleading as saying that it’s illegal to carry guns in Texas. In both cases there are applicable laws, but there is no blanket ban.

London has a knife crime problem in the important sense that any number of people being stabbed is a problem. However, it’s worth bearing in mind that cities like NYC have a slightly higher rate of fatal stabbings per capita. (Non-fatal robberies and assaults are tricky to compare across countries because of different data collection methodologies and different classifications.) Of course it would be good for fewer people to get stabbed, and knife crime is a serious problem for some specific communities, but the city as a whole is not experiencing the kind of knife crime epidemic that you might imagine if you get your news from alt right TikTok accounts.

Who are 'they'? There is no official thing called a 'knife ban', and again, there are no laws about knives specific to London. There aren't really any laws about anything that are specific to London, as there is no corresponding legislative 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