Surely, like airlines, you could pay to transport anything? Remember BR did basically run a delivery service - presumably you could transport a coffin like any other delivery.
Yes, until 1968 (I believe it was) BR was known as a Common Carrier, this meant that whilst others could and would refuse to take any goods offered, BR HAD to accept any kind of traffic for any destination.
This to some extent explains the Out of Gauge conditions that are still extant (although for different reasons these days) as any load had to be conveyed provided it could fit through the Structure gauge.
Common carrier status put BR at a severe disadvantage against competitors who would carry anything as long as it fited their vehicles, so we lost the cream and ended up with the sour.
The situation with coffins goes back to Victorian times and was related to the fact that there was not the road network available, even in the 30s/40s that there is now and thus to transport a coffin on the road network was not really a feasible option given the relative speed at which this could be done by train.
The Railway delivery services became National Carriers Ltd (NCL) and Brtish Road Services (BRS) the former delivering small packages locally from parcels depots, and the latter larger loads.
NCL's operations I recall subsequently became merged into the Red Star service as private carriers gained a foothold in the market. With the demise of parcels trains of course there was no network for them.