Bartleet wrong quite often. I definitely would not depend on Wikipedia for facts, but they were just one of many sources showing BSA stopped making complete bikes.
Bsa produced an early history leaflet in 1918:
The output of B.S.A. safety bicycles and tricycles steadily increased during the next three years, but in 1887 the Government demand for rifles and ammunition rendered it incumbent upon the Company to temporarily discontinue the manufacture of bicycles and tricycles, and for six years nothing was done in this direction beyond the production of a limited quantity of Kelsey's Duplex Safety Bicycle Crank Axle Bearings. During those six years the Company manufactured not only rifles, but also large numbers of six-pounder and three-pounder cartridges for quick-firing guns.
A bit ambiguous, but they mean that parts manufacture recommenced:
When the Government demand for rifles and ammunition somewhat declined, the Company once more was enabled to take up the manufacture of bicycles, and early in 1893 commenced the manufacture of safety bicycle hubs, examples of which were exhibited at the Crystal Palace Show that same year.
Remember that although they had stopped making bicycles in 1887, BSA components were still available from previous manufacture, so surviving bicycles (ordinaries) might exist with BSA parts.
It says that in 1908 they decided at a board meeting to start making complete bikes again. But they did not come onto the market in 1908.
In 1908, to meet the somewhat altered conditions of trade, it was considered advisable to manufacture complete B.S.A. Bicycles in the B.S.A. Factories. In announcing the new policy at the Annual Meeting the Chairman stated 'that the Company had decided to widen the policy in regard to the manufacture of complete bicycles. Hitherto only cycle agents were supplied with complete machines bearing an agent's transfer. In view of the somewhat altered conditions of the trade, it was considered advisable to manufacture a complete B.S.A. Bicycle, under the most rigid inspection, with the world-famed B.S.A. Fittings.
There would be one grade of machine-the highest possible quality only. The machine so offered to the public would be listed at a higher price than that at which bicycles built with B.S.A. Fittings had been sold by local agents and makers. The management believed that this would not only create a greater demand for B.S.A. productions, but that it would have a steadying effect on the cycle trade generally. Such a policy must tend to place the B.S.A. Fittings on a higher plane, and to a great extent prevent bicycles built with B.S.A. Fittings being sold at ridiculously low prices, showing little or no margin of profit to the retailer.' Although the new policy was severely criticised, events have proved that it not only helped the trade generally, but was much appreciated by the public. The complete B.S.A. Bicycle produced at The B.S.A. Company's works bears a special transfer.
The complete history leaflet reproduced here -
https://bsamuseum.wordpress.com/about/
Bear in mind that BSA Fittings provided customers with a much cheaper, but still top quality bicycle. BSA Fittings continued to be sold separately (alongside complete BSA bicycles), so Fittings bicycles continued to be made for many years after (particularly in Australia, where most manufacturers used BSA components)
Here's the 1900 BSA Fittings catalogue -
http://www.oldbike.eu/museum/1901-2/1900-bsa-fittings-catalogue/
Just to confuse matters, BSA did also make bicycle frames too in the early 20th century, which were BSA Spring-frames. These could be supplied By BSA as complete bicycles (in knock-down form). The idea of this was to provide the trade with a complete machine so they could fit an engine to create what was known at the time as a 'motor bicycle.' This was a particularly popular option in Australia.
A good example to understand the nature of a BSA Fittings Machine is this bike, which has a small local builder's decals, but is totally a BSA, i.e. all components plus the spring-frame made by BSA -
http://www.oldbike.eu/museum/1902-2/1904-bsa-spring-frame-roadster/