Could HST sets be modfied internally to take Royal Mail (or any of their competitors, who get much of the corporate work) cages and/or bags? Wouldn't need much change. They could get them on with a ramp like they do wheelchairs, something to hold them in place. They could leave them in the sidings 10/11 months of the year and roll them out for christmas. They would be available for the rest of the year if neccessary. Might even be a bit of a trojan horse for mail on rail, winning back routes one at a time.
Mail sorted for the midlands could get the train out late from St Pancras, Yorkshire etc. from KX, points west from Paddington. The only competitor for speed would be planes, which would need a slow lorry through the traffic to the airport, while rail delivers it to waiting vans in the city centre. No airside passes, no vetting drivers to CTC level, no congestion.
Derby, for example, could put their stuff going further north on when the Derby post was unloaded and their stuff going south on the equivalent running to London.
It wouldn't be any heavier than a packed passenger load once you've taken out the seats and toilets, so no braking, speed or acceleration issues.
It could even be shared, with DHL taking a carriage, TNT another, Royal Mail 1 or 2, maybe even the operator taking one for all the smaller firms. Special Delivery (and any really expensive parcels as well as the sensitive DX stuff) could go in the toilets with a new door that locks from the outside. Government Mail might take a carriage here and there for trunk stuff. They could use the toilets for red boxes which now commonly travel in empty (Government Car and Despatch Agency) cars to their ministers.