As we're talking about overhauls - I have my ideas; though absolutely not fully thought through...
I'd Federalise England into 4 regions, with their own devolved parliaments, in Wessex (Bristol / Winchester), Mercia + E.Anglia (Birmingham / Tamworth), Northumbria (Manchester / Durham) and London (bound by M25).
Those 4 plus the 3 Celtic parliaments all to have the same powers as each other (notably more than Hollyrood currently has).
NZ-style compromise of constituent and PR voting for the devolved parliaments.
Those devolved parliaments nominate from within themselves (proportionately) to Westminster and a smaller, national HoC; government explicitly not made up from 1 party.
Revamp the HoL to be the elite / sinecure for the highest representatives of their trade - to serve 1 lengthy term. HoL acts as advisors, as well as check and balance on the government of the day. For tradition's sake, I'm happy enough to call them all Lords and Ladies (and to keep the ***le after their term is up - but no longer sit there, and not to hand it down to their kids). So yes, HoL will still have the odd politician, bishop and celebrity, but an equal number of top barristers, accountants, generals, teachers and medics etc.
The explicit purpose of the HoL would be their collection of expertise covering every field, and they should mostly be there to advise the HoC; and only to shoot down unconstitutional* legislation.
*Yes, I'd have a written constitution; explicitly to be revised every 100 years or so; and explicitly requiring national referenda on any changes - dealt with much more like the Good Friday Agreement, than the Brexit "debate".