If you go far enough back, you find a Manchester-Edinburgh via Hellifield, Carlisle and Galashiels (the L&Y/Midland/NB route).
Possibly with a portion from Liverpool Exchange attached at Blackburn.
Really, it all goes back to the agreements between the pre-grouping railways, and the way grouping was done in 1923.
The Liverpool-Newcastle was a joint LNWR/NER service, but the LNWR wouldn't have had an arrangement with the North British as it was deeply tied in to the Caledonian.
So no trains to Edinburgh via the NB.
I was fascinated to learn that the LMS had a 30% ownership of the Forth Bridge.
It got this via the Midland, which put up 30% of the original cost, with the ECML companies (GN/NE/NB) having the major share.
So the Midland worked with the NB (and G&SW), while the LNWR didn't.
Even in the 1960s these company niceties were still being followed by BR.
Manchester-Scotland had one train with portions from LNW stations (Man Exchange, Liv Lime St via Wigan NW), while another used L&Y stations (Man Vic, Liv Exchange).