Not necessarily; The Great Central, for instance, remained with the E.R. for many years, even though it started in Manchester and Finished in London. I don't think the LMR completely took over the running of the CLC until 1952.
On another note, a well-known model shop proprietor in Saddleworth once told me that on the on Standedge route Newcastle - Liverpool trains, the Ex-LNER stock could still be seen in teak as late as 1953. There's another thing; there were two different routes for these trains, of long standing. One service was ex-LYR/NER, and ran from Liverpool Exchange via Rochdale and the Calder valley; and the other ex-LNWR/NER, which ran from Lime Street, via Standedge and the Colne Valley. Did this one run via the New Line from Heaton Lodge to Farnley Junctions? Certainly on the Standedge route the LMS and LNER provided complete sets of their stock on alternate days, for the coupling and gangway connection reasons mentioned above.