Just to make clear a couple of definitions:
Passenger trains are any train that is supposed to convey passenger. They may also convey parcels, mail, newspapers, fish, live chicks and so on in the van, but they are still a passenger train. They may have various vans in the formation or on the back: if these are what were latterly called NPCCS (Non-Passenger Carrying Coaching Stock) they are built to standards that allow them to run in passenger trains and their inclusion does not in any way alter the situation if the train is intended to carry passengers.
This is true - when I mentioned mail or newspaper trains conveying a small amount of passenger accommodation being "a sort of category by itself" I mean a topic of discussion rather than a recognised category of train - they would be classed as passenger trains as you say if the seating accommodation was for public use.
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Going back to the original question and interpreting it broadly to include all kinds of newpapers / mail / parcels / sundries on passenger services, there would be a few different dates when different kinds of traffic ended.
End of "mixed" services which conveyed passenger coaches and freight wagons - last service appears to have been the Mallaig oil tanks, ended Jan 1989 or possibly late 1988.
End of "sundries" traffic on passenger trains that was being forwarded by BR (rather than miscellaneous items being sent through the mail)
End of passenger services on which the majority of vehicles were NPCCS (i.e. advertised passenger services with a majority of mail/newspaper/parcels vans and limited passenger accommodation)
End of newspapers being conveyed on passenger trains
End of general parcels traffic being conveyed on passenger trains
End of mail being conveyed on passenger trains
End of Red Star parcels being conveyed on passenger trains
Last use of a passenger station to load or unload Royal Mail
N.B. I'm not quite sure where some of these categories begin or end, for example Royal Mail traffic includes a certain amount of parcels but there was a separate "parcels" business from what I understand.
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Newspaper traffic on BR ended in July 1988, with a rapid decline setting in from 1986 with Murdoch's move to Wapping mentioned above in post #32, so passenger trains conveying newspapers presumably ended during this period* (unless you count things like free newspapers handed out in first class!)
There's a long article on newspaper traffic on the Railway Magazine website - two key sections are quoted below.
The preponderance of (usually grimy) vans meant that it was not always easy for a lineside observer to differentiate between a newspaper working and an ordinary parcels train, but in the case of some of the longer-distance corridor formations, it was occasionally possible to do so by the presence of a BSK or similar seating coach marshalled into the rake. This would not only provide toilet facilities in addition to those found in some of the vans, but would allow the staff to relax and get some sleep if they finished their duties before the end of the journey. It wasn’t unknown for such carriages to provide an unofficial early morning Sunday service for late-night revellers who’d missed their last Saturday night train… the staff usually turning a blind eye if the service wasn’t one of those advertised in the passenger timetable.
The first blow to the newspaper train industry fell in the 1980s when The Guardian decided not to renew its contract with BR and switched to road transport, but by far the biggest nail in the coffin was the move of Rupert Murdoch’s News International (publishers of The Sun, The Times and News of the World) from Fleet Street to new-tech printing premises at Wapping in 1986. Murdoch knew the rail unions were threatening to ‘black’ papers printed in Wapping in protest at print job losses and had therefore signed advance contracts with road hauliers for the dispatch of papers in articulated lorries. No longer would those market-leading titles find their way to readers via the London stations. That move took a large chunk out of BR’s £125million-a-year paper and magazine distribution business – and matters were to get worse the following year when it lost another big contract with Mirror Group Newspapers. A number of nightly trains continued to run for the remaining traffic, but with the income continuing to haemorrhage, BR undertook an assessment and concluded that the business was no longer worth pursuing. Overnight sorting of newspapers in dedicated trains thus officially came to an end on July 9/10, 1988.
https://www.railwaymagazine.co.uk/6311/newspaper-trains-read-all-about-it/
There's also this thread on newspaper trains, with a list of 1970s workings, some of which ran as passenger trains:
https://www.railforums.co.uk/threads/newspaper-trains.31425/
In a thread on RMweb, there's a suggestion that the carriage of mail on passenger trains had ended by 1996, with the new Royal Mail contract signed that year being purely between Royal Mail and EWS.
https://www.rmweb.co.uk/forums/topic/170460-the-last-mail-on-a-normal-passenger-train/
The situation with Red Star parcels is summed up on Wikipedia as follows - I assume the carriage of Red Star parcels by passenger train ceased when the station-based offices closed?
On 25 May 2001, Lynx Express decided to shut down all of Red Star's station-based offices and attributed this decision had been taken in response to the substantial disruption across the rail network in the aftermath of the Hatfield rail crash of October 2000.
https://en.wikipedia.org/wiki/Red_Star_Parcels
Not really part of the original question, but the final separation between the mail network and the passenger network must have been in 2004 when the Royal Mail contract with EWS ended. Up to this point, although the "Railnet" system mainly used dedicated terminals (such as Willesden PRDC, Shieldmuir and so on) there were still some passenger stations - although not passenger trains - which handled mail, such as Swansea and York.
This is illustrated by a few photos:
Royal mail lorries at Carlisle station in 1998, with a Class 325 mail unit in the platform behind:
https://www.flickr.com/photos/75784477@N08/14333880629/ (Photo: Dave McDigital)
The last TPO working to call at York, with 90027 in January 2004:
https://www.flickr.com/photos/phil_mason/21390462228/ (Photo: Phil Mason)
Penzance TPO at Exeter behind 67029, later the same month:
https://www.flickr.com/photos/33949742@N04/51839842228/ (Photo: BigSky47539)
From memory, I think all mail after that (when a slimmed down network of Royal Mail trains resumed with class 325 units) was only handled at the dedicated Royal Mail hubs. One possible exception is Stafford Royal Mail platform, which was a dedicated mail platform at a passenger station. It was regularly visited by class 325 units on test from Crewe for many years, but I can't remember when it stopped being used for mail.
There have been examples of miscellaneous items / parcels conveyed by passenger train since then. Fish on the Caledonian sleeper was mentioned in post #13, and there's a thread about lobsters and shellfish by passenger train from Cornwall to London in 2015:
https://www.railforums.co.uk/threads/gwr-delivers-shellfish-by-train.123844/
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*Edit - Reading through the excellent Railway Magazine article on newspaper trains more thoroughly, it mentions that after the end of dedicated newspaper trains in July 1988, newspapers continued to be moved on passenger trains (on a much smaller scale) until privatisation:
A handful of vans remained in use in the south for a few weeks, but they had all gone by November. After that, a few small supplies – tiny in comparison to the huge bulk loads of the past – continued to be conveyed in the guard’s vans of sleeping car trains and ordinary passenger services, but by the time rail privatisation came sweeping in like a new broom in the early-1990s, it was all over.