Well, as an east midlands resident with a partner who is keen on Stilton I'd say that was quite probably the case, although maybe the time the local diary had the contract did not precisely overlap with your grandfather's time there.
Apols if you know already but Stilton has to be made in a confined geographic area and Melton Mowbray is towards the south of said area I believe. Many dairies around the area would have existed then I would think, and produced Stilton Cheese, no doubt at least one had the contract with the House of Lords catering department I suspect. They would have taken it to a local station for obvious reasons.
I think there are now only a few of the smaller trad diaries in the area that make Stilton (with larger diaries sited in the permitted production area for the large scale market eg the supermarket shelves I guess)
2 of the smaller scale ones that still exist in the area (and we've been to buy stilton at the dairy in Colston Basset) are
Colston Bassett - nearer Harby further up the same line
and
Long Clawson - fairly close to Scalford or Old Dalby on the MR line (now the ex BR test track)
https://www.colstonbassettdairy.co.uk/
https://www.clawson.co.uk/our-story/NW Joint once more; and Long Clawson:
Sadly neither mention railway transport of product, but of course if they sold beyond the immediate locality they would have to have used it.
You post makes me think that during these lock down days I'd better get some stocks of stilton from our local shop!