A BBC Micro would strike me as an over-engineered solution (from when it was available). Even if you went for the cheaper Model A, you'd have a high-speed processor, custom video chip, and far more (expensive) RAM than is needed to drive a teletext-style display. (1K would do for the display RAM). And if using an off-the-shelf Beeb, you'd need a separate machine for each platform.
The teletext-style display (the Beeb's MODE 7) could be implemented with three chips: 6845 video address generator/CRT controller, SAA5050 teletext generator, and a 1K RAM chip. (The Beeb used these very chips to generate its teletext mode display!) You could then use whatever microprocessor system you wanted to modify the contents of that RAM chip, in order to update the information displayed on the departure screen.
It'd be very straight-forward to use one microprocessor to update the display RAM used by several departure screens. This could be an integrated, bespoke system, or a Beeb with an expansion card with 6845/SAA5050s for each connected screen.
The above corresponds to my recollections of colour departure screens on CRT TVs in the 90s. But other posters have stated that black and white screens were more common for many years. That makes perfect sense, given that monochrome CRTs were far cheaper to make/buy than colour for many decades.
But if you're designing a system to drive a monochrome CRT, then the colour-output of the SAA5050 would be largely wasted. All you'd be left with is the ROM used to map display memory contents into the correct scanlines for each character, and the ability to make some characters flash automatically.
If all you want is a character ROM, you could use any suitably programmed ROM chip for that purpose, combined with a shift register and a few more components to make a serial data signal to drive the CRT's brightness as the electron gun scans across the screen. As an added benefit, you can program the ROM chip to have whatever characters (or fonts) you wish.
So now we're down to a video address generator (e.g. 6845), a custom-programmed ROM chip, and a little bit of RAM. That, plus whatever microprocessor system updated the display RAM, and perhaps a communication system to obtain that data remotely.
As for when these devices would have become available, the microprocessor revolution really got under way - and affordable - in the mid-1970s. The other hardware (CRT controller, ROM, RAM) were also available from then, and got cheaper year-by-year.