Fair to say:
"Country" is a somewhat colloquial term. "State" or "sovereign state" are more legally precise terms, and country is generally used as a synonym thereof.
The UK is undeniably a sovereign state, and the home nations are defintively not.
England and Scotland were sovereign states for centuries, and Wales was briefly, too (NI never has been).
Pick up a map of the world, which shows its countries, and it will almost certainly show the UK by its borders, but not the constituent nations. Ditto, pick up any encyclopedia, or read Parliament publications, which will confirm that the UK is indeed a country. There are only relatively few international contexts (e.g. certain sporting evenets) where the 4 nations rather than the UK are treated as countries - and that's usually because of the historic British role in the establishment of the sport.
Can the home nations be called "countries" too? Sure, but they aren't countries in the ordinary sense of the world, particularly outside of the UK.
If the USA rebranded to UCA - making no constitutional changes but just using a different word - would you view it as 1 country or 50 countries?
Because that's basically the difference. Brits (and a few other states) use the term "constituents country" to identify its historic geographic regions, where other countries use different words (state, province or territory) to mean functionally similar things.
In any event, the UK is a country within any typical usage of the word.
This isn't a reflection of my personal, political beliefs - I support Welsh independence - but just of the present legal/linguistic reality.