This actually requires a pre-1900 PoD. And historically speaking the four Home Nations have been a unified part of the United Kingdom, all four are historically separate. The (English) FA is the oldest FA in the world if I remember correctly, and initially based on a merger between a London "FA" and the Sheffield "FA" forming the English FA. Scottish FA popped up as an agreement between Scottish clubs; even in the 1800s, there was recognition that Scotland has it's own identity and way of doing things, and likewise for Ireland (later reduced to Northern Ireland) - these countries had not been unified with England for that long in the grand scheme of things.
You then had the Home Nations form IFAB (which continues to maintain the official rule book) as the body for organising internationals, including England vs Scotland despite both being within the UK. The later creation of FIFA, and "grandfathering rights" allowing the Home Nations to join was merely a recognition of what was already happening.
To be honest, I think the separate home nation teams are a result of the nature of the UK as a "country of countries", a situation which seems to be quite rare in the world as most other comparable examples have broken up or had local identities erased. I don't really see how it's possible to avoid this situation without making the UK more unified to try and erase national differences. The UK Government has normally avoided interfering in private affairs, which is how the national FAs started out.
FYI; I don't think Celtic and Rangers were ever offered (or even partook in actual discussions) over joining the English league structure. Celtic and Rangers have normally looked with a jealous eye at the money on offer in the Premier League. However, there is divided opinion (to be diplomatic) as to whether they are on a PL level currently, or whether the Championship would be an appropriate level to enter in to. Given time and the PL money, they'd probably establish themselves as PL teams, but currently personally I think they'd struggle. They can compete with PL teams in the Champions League on an off basis, but to use an over-used phrase, I think they'd seriously struggle to do it every "cold Tuesday night at Stoke".