Also if in this ATL the UK adopts a written constitution in the late 19th century or early 20th century, is it plausible that the Church of England would be disestablished? I know that in the latter half of the 19th century many Non-Conformist Protestants were in favour of this and I would assume that there would have been significant support for this in the Liberal Party and later on the Labour Party.
Yes, but there was antidisestablishmentarianism as well.
OTL, Anglican Church was disestablished in Ireland in 1871, and in Wales in 1920, but not in England. So, a written constitution in that timeframe would probably explicitly establish the Church.
If this written constitution is put in place in the 19th century, then it would presumably give the Lords quite considerable power.
To some extent, given that for a written constitution to be put in place, there would have to be a more radical political environment in the UK at the time, the Lords may to some extent be democratised in a limited fashion. However it will still have more powers and given that written constitutions are usually designed to be hard to change, we could in this ATL have a House of Lords that is today more powerful than it is in OTL.
Sure. And what about Lords who are elected by people? They would have a democratic mandate to compete with the MPs in Commons, more than the hereditary lords.
In OTL, Lords lost their final veto in Parliament Act of 1919, when the Commons term was limited from 7 to 5 years, and Commons could not extend their term without permission of Lords. In 1958, life peers were introduced to pack the Lords.
There were elected Lords in the united Parliament, 16 representatives of Scottish peers and 28 representatives of Irish peers.
Suppose that sometime end of 19th or early 20th century, the Lords were reformes so that the 16 representatives of Scottish peers and 28 representatives of Irish peers remain, and the other peers get to elect their representatives - like 100 or so. And the majority of House of Lords is filled by popular election - for life. In this case, the new Life Peers have direct popular mandate, and they also are a different set of people than MPs in Commons because Commons are elected for 5 years, and Rerpresentative Peers for life.
What next?