Quite a lot to ask in one go! Some of which is searchable, but what the heck...
AFAIR, there are about 2,500 ATCOs in the UK, of whom about half are employed by NATS, and the rest by the RAF, RN and non-NATS ATS providers such as Serco and those airports which run ATC themselves. Not all ATCOs are operational - I would expect that number to include a good number of managers, project staff, instructors, etc, who hold ATC licenses.
The shortage of controllers is probably due to a variety of factors. While there may be a small retirement bubble effect, it is almost certainly much les significant than in, say, the US or Canada where recruitment can be halted for years at a time. Other factors are a small training capacity, fairly low success rates in training and some attrition to overseas. From to NATS perspective, there are also losses of controllers to other domestic ATS providers though NATS is more likely these days to have a net influx of ATCOs from other UK employers. In the old days, a lot of people who failed the final courses at CATC would have aerodrome ratings and thus be binned by NATS and picked up by non-NATS airports, eventually returning to NATS in many cases as experienced ATCOs.
Working abroad is an option though not one to shout about before you have a few years of valid experience. Eurocontrol doubtless has some ex-NATS ATCOs as well as British citizens trained by Eurocontrol and outwith the usual British training and employment stream. The number of British ATCOs working in the Middle East is probably in double figures, so not a hugely significant number. A low double figure are in Canada and there must be some in NZ and Australia, and perhaps a very few in South Africa. There are isolated pockets elsewhere, such as with DFS in Germany and Skyguide in Switzerland.
The shortage of controllers is a worldwide phenomenon, thanks in large part to underfunding of recruitment and training programs and the preference among many employers to run short-staffed and save a few bob. Given the overtime bills in some locations, that can be a double-edged sword for the ATS provider as well as relying hugely on the goodwill and professionalism of staff. If we all work to rule, the system will break! Added to that, plenty of countries don't do their own initial ATCO training or rely on expats.
Personally, I'm glad to work somewhere where overtime is unusual and we have time to recharge between cycles. It shouldn't be long (though controllers can be their own worst enemies in this regard) before there is a day of reckoning in those countries with monopoly ATS providers trying to get away with understaffing. Certain countries, thinking perhaps of those which produce such things as luxury chocolate, boomerangs and hockey pucks, would grind to a halt without overtime. It's about time for ATCOs to refuse overtime and refuse to work beyond safe traffic levels, and make a good
PR case rather than allowing some ATS providers to use the line of "there's no sympathy for highly-paid ATCOs". As soon as you ask the travelling public how much they'd like you to be paid, and how much time off you should get, when they and their loved ones are flying, you find a LOT of sympathy...