Was it just a loss of face or pride thing?
There's also a weird strain of Protestant Unionism that is stuck in the 18th century but seems just below the surface (and on the surface when the Orange Order are marching)
Nonetheless I think in general, as a UK citizen myself, that the British (ie UK citizens, the nomenclature gets convoluted!) consider the Union to be central to UK strength as a nation in terms of society and business and much else (education, sport, language, identity, history, position on the world stage, defence, ...).
Dissent grows though - once the Pandora's Box of the West Lothian Question had been opened and it started to sink in, and we saw that many in Scotland wanted to ditch the rest of the UK and make a run with what they perceived to be vital resources, that I feel rather unsettled people. Why shouldn't the rUK have voted at the same time whether to kick Scotland out of the Union too - that would have only been fair.
Her Majesties government certainly made efforts to retain Scotland in the Union, that could well be key.
I can see that there is an impact in defence - Trident is based here and the Scots have always been over-represented in the UK military. The rest just sound like potentially damaged pride to me...
"consider the Union to be central to UK strength as a nation"
Really? Scotland is a small and relatively unimportant part of the UK. What possible direct impact could our gaining independence have on the rUK apart from damaged pride in that whoever is PM at the time would get made fun of at the next G8 summit for losing part of his/her country?
I'm British and have lived part of my life in Scotland, I don't consider it another country, it's part of our country and my relatives who live there are the same nationality as me.
It's a visceral, and probably illogical, thing to a certain extent - like feeling attached more to family members than to other people.
Also Scotland ain't that small, it's population is that of Yorkshire, it's geographic size is pretty big as a proportion of the UK (~1/3 I think).