Interesting rapportage between you all ... The board on show in the Museum is cream with brown lettering and a brown edge. The reverse is white with unusual green lettering for University of somewhere...
We haven't really addressed your question very much, just gone off on our own tangents. I'm afraid we're a bit like that !
The BR carriage livery from 1950 was red and cream, and these long boards were done in the same colours. In 1956 the standard livery changed to allover maroon red, and the boards were now done maroon, with yellow lettering. However, the Western Region was "permitted" to paint it's named trains (only) in the traditional GWR chocolate and cream, with destination boards painted to match. There was great keen-ness among former GWR staff about this, and apart from the established named trains a whole series of new ones were invented, typically one, the best of the day, on every main route. I bet the painter at Swindon still had plenty of cream and brown paint left over from the old days. Now we know your board is brown/cream we can say it is from the Torbay Express, which was a daily service from London Paddington to Paignton/Kingswear.
Boards were difficult to change, as you can imagine, and normally left in situ. They were put on/off in the carriage depot using the carriage cleaners' staging and a couple of staff. You can't really get them up from platform level without equipment. The small clips at either end to drop then into were standard features on all the main line rolling stock of the era. The reverse was normally plain brown, so if the coaches were to be used on a different service they could just be reversed, although the same specially painted vehicles normally stayed on the service long term.. It looks like someone has made use of the blank side of yours for some later purpose.
The chocolate-and-cream express coaches only made one journey a day, returning next day, so two train sets were required for the daily service, passing en route. This was felt uneconomic in the Beeching era, so it was determined to end the special livery and make all trains have the maroon livery, and they could cover multiple services all day. So in 1963 this livery was abandoned. Although it took years to repaint them all, the dedicated coloured boards were given up pretty straight away. However the Western Region didn't want to let them go altogether, so they moved on to the waist-level small boards described above, which could be changed by platform staff. These lasted to about 1970.
It just occurs as I write, but the reverse side is not the University of Exeter, is it? Not only local to you, but the University had a railway society who in the 1960s-70s used to charter a full train at the start and end of term to Paddington, and sell tickets for it to students all around the university. They more than once wrote an article about it in the rail magazines of the time. I wonder if someone from there managed to get hold of one or more of these boards when they were no longer in use, and painted them up for their special train.
Given that your board is from the Torbay Express chocolate-and-cream era, I very likely used to see it pass me in my youth when I spent school holidays etc watching the trains pass at Taunton. The Down Torbay used to pass at lunchtime, it would hammer through at about 80mph with those distinctively coloured vehicles, quite nicely cleaned I recall. So I'm glad that some little bit of what I used to see long ago has found a good home.
Meanwhile, move on a few years and by 1966 we had moved away from Somerset to the north of England. However, we took a family summer holiday to Cornwall using the Motorail service, along with our car, that ran from Newton-le-Willows to Newton Abbot. You'll find one or two mentions of this trip by me elsewhere here. Now at Newton Abbot, on the return, there was a railway relics shop in those times at the north end of the Up main platform, which (of course) we visited when our car had been loaded while waiting for the return trip. There I obtained several things like old single line tokens, which I still have. But in there, taking up much of the space, were a number of those big high level destination boards. Now how could I get those, being about 15 feet long, home? I suggested we could place them on the car transporter wagon alongside the car
Alas no, so they were left there. I do wonder if nobody else managed to buy them either, and it's the same ones that have ended up in your museum in the town.