Pigeons and the UK Law

Feral pigeons are considered wild for the purpose of the Wildlife and Countryside Act 1981, which declares that intentionally killing, harming or disturbing any wild birds, including their nests or eggs, is a crime.

There is no legal definition of 'vermin'.

A common excuse people often give when trying to get you in trouble for feeding pigeons is that dropping birdseed is considered 'littering'. According to section 87(1) of the Environmental Protection Act 1990, "A person is guilty of an offence if he throws down, drops or otherwise deposits any litter […] and leaves it." So until you walk away from any uneaten birdseed, it is not legally considered littering, and the birds are more often than not quick to tidy it all up on their own. Provide food incrementally, stop if the birds seem to become uninterested, and don't leave until it's gone if someone's pestering you about it. Uneaten food left around can attract unwanted attention, and harm the animals that consume it if left to spoil, so it's important not to offer more than the birds can finish in a reasonable timeframe, anyway. As for "attracting vermin", I think if any authority actually wants to tackle that as an issue instead of just wanting to look like it, people feeding birds are going to be much lower on the list of priorities than those responsible for managing restaurant waste and overflowing public bins.

Sometimes signs go up warning the public against feeding the birds because of a high risk of bird flu... and then don't come down again, even once the hazard level in the area has reduced. In the UK, you can check the current status and restrictions of your area on the APHA's interactive map. It's important to stay updated to make sure you're not doing the birds (and the public) more harm than good. There are also official government updates to be found here [England|Scotland|Wales|Northern Ireland].