What laws are there against paparazzi in the UK. Do they have to leave celebrities alone when they are told to? Are they allowed to hang around outside the celebrity’s house? At what point does it become stalking or harassment?

If they are in a public place then they cannot stop you from photographing them (their security might try though),if they are only in the grounds of their own house or someone else's then you can't as that is a crime as it is private property.

Not that that stops the scum from trying

Same laws as for anyone else. You cannot trespass. That’s about it.

What did change in the UK is the national policy that news journalists should follow. As to paparazzi-style photography:

“i) Journalists must not engage in intimidation, harassment, or persistent pursuit. ii) They must not persist in questioning, telephoning, pursuing, or photographing individuals once asked to desist; nor remain on their property when asked to leave and must not follow them. If requested, they must identify themselves and whom they represent.” — Independent Press Standards Organization
https://www.bustle.com/p/how-have-paparazzi-laws-changed-since-princess-dianas-death-the-tragedy-is-still-affecting-policy-72504
How Princess Diana's Death Changed The Paparazzi