Above Paul Theroff said:
And now that there is virtually no constitutional significance to hereditary peerages, I expect that soon the House of Lords itself, the Lord Chancellor, Attorney General, etc., will be removed from involvement in adjudicating peerage claims, certifying newly inheriting peers, etc. There is now certainly no reason why any public officer or public expense should be involved in deciding such matters.
The House of Lords (Hereditary Peers) Bill does explicitly remove the jurisdiction of the House of Lords from adjudicating peerage claims.
In the immediate future I assume the Lord Chancellor / Secretary of State for Justice will continue to admit peers to the Roll of the Peerage in line with the
Royal Warrant of 2004. It was discussed in debate that it is the government's position that any contested claims would be heard by a committee of the Privy Council (as they are for baronetcies).
Similarly there is a Roll of the Baronetage maintained by the Lord Chancellor under a 1910 Royal Warrant, and they have never held a position in the Constitution; and in Scotland there is the Lyon Office which is a publicly funded office for some hundreds of years and that seems to deal with heraldry and Clan Chiefs, which again are not constitutional functions.
Of course, nothing lasts forever, and I can definitely foresee a time when the state simply washes its hands of recognising or not hereditary peers / baronets / etc., but I think this is still a few decades off.