The privilege of peerage in criminal proceedings was abolished by s. 30 of the Criminal Justice Act 1948.
As for why, see Colin Rhys Lovell, "The Trial of Peers in Great Britain", The American Historical Review, Vol. 55, No. 1 (Oct., 1949). There were "grave disadvantages in the procedure to the accused peer, who could not waive it." The accused could not challenge the jurors or appeal their decisions. It was ultimately decided that the procedure had "outlived its usefulness."
See also some justification from Hansard:
[...a peer] cannot say, "Waive all this, and treat me like anybody else." The existing law is that he must be tried by this grotesque and antiquated arrangement. It is a Court, but it is a very ancient Court, and it is high time that provisions were made that a Peer who is charged with a felony should be treated in the same way as anybody else.