It is worth clarifying that althougt they are covered by the Copyright, Designs and Patents Act (and its subsequent amending regulations) what you are talking here are two separate things: Copyright (in the words and music) and Performance Rights which lead onto Recording Rights. The performer(s) generally own the right to exploit their performance (either in the studio or live on stage) through recordings, ticket sales etc, and the record company generally owns the right to exploit the recording, in record sales, airplay and public performance of the recorded music (eg a DJ or a funfair providing music as background) The performance right and the recording right are known as economic rights and are covered in Part 2 of the Act.
There's no one standard answer to your question, as each record company may have a different policy with regard to its contracts. And a well established band with more clout may well have a diffrerent contract to a newly signed band.
In most cases the record label will insist that the band signs over their rights in performance of the songs - usually the recorded (studio) performance but this may also include live gigs - but without any copyright in the lyrics or music being affected. In exchange the band will probably receive an advance and royalties based record sales and airplay revenues etc. Session musicians are usually paid a flat fee and sign over their performance right without getting royalties, although this is not always the case.
As some bands nowadays can make more money from touring and live gigs than they do from record sales, the record companies are trying to muscle in on the live performance revenues which previously would have been the province of just the band, their management and the promoters or individual venues.
So you would need to check with the band's management in each case. It would be sensible to make sure that when you get the written permission to record the performances this includes a statement indemnifying you from liability for any infringement of the performance right of any third party. This won't entirely protect you, but it will provide you with a defence under Section 182(3) which says:
So in other words if the band have previously signed over their performance right to a record company, and you are held to have infringed that right not knowing this fact, you will not have to pay damages, although you may have other sanctions applied against you. The most serious of these would be that you would have to surrender your infringiing video recordings."(3) In an action for infringement of a performer’s rights brought by virtue of this section damages shall not be awarded against a defendant who shows that at the time of the infringement he believed on reasonable grounds that consent had been given."