I first heard about this push for independence in Pat Buchanan's book State of Emergency, citing the increasing prevalence of separatism throughout the world: Quebec from Canada, provinces threatening to separate from Brazil, Czech Republic's split from Slovakia, provinces of the former Yugoslavia, the continued fragmentation of the former Soviet republics, and increased bloody wars and tribalism worldwide.
In Scotland, the desire for a split is attributed to the British loss of empire. When Britain was Great, Scotland had more desire to be a part of it. With the loss of prestige and empire, old differences and hostilities have risen, and a greater desire to go their separate way.
It was Slovakia who split from Czechia, actually. While Slovaks and Czechs see each other as brothers due to similar cultures, languages, and historical backgrounds, Slovaks got tired during the post-war years of communism over how Czechs dominated Slovaks in various ways. Not just economically and politically, but also culturally. If a Slovak got famous abroad, for example for being an excellent ice hockey player, Western media would report about him being a Czech.
When communism fell, Slovaks saw the chance of independence. Luckily for all parts, this got solved peacefully (unlike Yugoslavia).