Hi Charlotte,
I don't see any other way unless the funeral train was routed from the Ilkley line onto the Halifax line. I think that it might even have involved a change of station in Bradford

. The rail network was far bigger then than now, it might even have been possible to go the other way round i.e. Ilkley, Skipton, Keighley, Halifax, Bradford.
One alternative would have been a train from Ilkley to Bradford then a horse drawn hearse travelling the 5 miles or so to Scholemoor with the mourners following on foot or perhaps in horse drawn 'charavans' .
There are wide aisles at Scholemoor so carriages, hearses and so on would be able to drive right to the grave side.
There was a web site dedicated to 'underground Brittain' where there was a lot of older railway information, try Google it.
Bradford T&A as well as the Observer might have reported the funeral, if so then the staff at Bradford Library might be able to help you.
Dave