The first bullpup to see major service use was the Austrian Steyr AUG (AUG = Armee Universal Gewehr = universal army rifle), which was selected as the standard infantry rifle for the Austrian army in 1977. This rifle was later adopted by Australia and New Zealand, and also by others. Australia, aiming to switch to a 5.56mm weapon (from its 7.62mm L1A1, a variant of the Belgian FN-FAL), tendered for a new rifle in 1983, and ran trials to choose between the Steyr and the M16A2 in 1984-1985, selecting the Steyr. The Australian version was a variant, made locally under license, as the F88 or Austeyr, seeing issue in 1988. New Zealand beat Australia to using a bullpup, importing the Austrian version in 1987, and later the Australian F88. One Australian weapons technician involved in the Australian trials commented on the Steyr:

It didn't look sexy. It was green; it didn't look like a weapon; it didn't look like a real gun. It was something that looked like a toy. That can’t be a gun. Guns are black and they're made out of steel and rugged stuff, not this crap.

The Steyr featured in The Running Man (1987) as a futuristic science-fiction weapon.

The British also replaced the 7.62mm L1A1 with bullpup at the same time, adopting the 5.56mm SA80 in 1987.

The Steyr only just beat the French 5.56mm FAMAS bullpup into service. The FAMAS was an older design, already in prototype in 1971, and being tested by the French army in 1972. However, they were slow to officially adopt it, only doing so in 1978.

Bullpups have a much older history, though. The British were working towards a bullpup at the start of the 1950s, evaluating the EM-1 and EM-2 rifles (EM = Experimental Model; the EM-3 and EM-4 never got off paper), both bullpups with intermediate power cartridges. The EM-2 was selected, and officially issued, but only got to the troops in small numbers before the NATO standardisation on the full-power 7.62×51mm rifle cartridge sank the design. The British, like much of NATO, adopted a version of the Belgian FN-FAL (the German army adopted the G3, a version of the Spanish (designed by a German) CETME, and the USA adopted the M-14).

There were older bullpup designs, such as the Thorneycroft carbine of 1901, which didn't see serious consideration as military rifles.

While bullpups have the advantage of being able to provide a longer barrel in an overall shorter length, the design does have some potential disadvantages. For example, the chamber and magazine are below the user's line of sight, and checking them or clearing blockages means that the use must look down - this reduces "awareness of the battlespace", as they say. While not strictly a bullpup problem, bullpup rifles are mostly modern, and designed to be cheap and light, and parts sometimes turn out to be too fragile in the field. New Zealand recently replaced the Steyr with a non-bullpup (beginning in 2015, and completing the process this year). Part of this is due to recent fighting by New Zealand soldiers in Afghanistan showing that battle often takes place at longer ranges than the 300m or less that the Steyr was designed for. Part of it appears to simply be dissatisfaction with the Steyr:

It is a highly overrated assault rifle and if given the choice, I would rather throw stones at the enemy than carry that stoppage prone piece of crap

References and reading:

u/handsomeboh already noted the Thorneycroft and the EM-1/EM-2, and the Steyuyr; I have merely expanded on this here.

The Thorneycroft patent: https://worldwide.espacenet.com/patent/search/family/032423717/publication/GB190114622A?q=pn%3DGB190114622A

The Australian selection of the Steyr: https://www.aspi.org.au/report/sticking-our-guns-troubled-past-produces-superb-weapon

New Zealand ex-soldier quote from: https://archive.is/20140620051915/http://www.nzherald.co.nz/nz/news/article.cfm?c_id=1&objectid=11278332