Hi bobshop,
BA don't operate point to point services between destinations in Europe. What your family might be thinking of is the likes of Deutsch BA, who are the German 'branch' of BA (soon to be painted a nice shade of easy orange, if easyjet complete their option to buy the company) and do operate domestic routes in Germany, as well as from Germany to Europe - specifically to places like Malaga, and the odd flight to Turkey as well I believe. Mostly though it's German domestic routes. BA also have a Scandinavian branch, they used to be called Sun Air, and operate ATP's and Jetstreams around Scandanavia and also to the UK on behalf of BA.
So in a way, yes, BA do have aircraft flying around Europe, but they are not mainline BA aircraft- they are operated by other companies, often in the BA livery. Until recently, Air Liberte was another example of this, in France. It's now been rebranded as Air Lib, and as far as I know isn't assosciated with BA any longer, although I may well be wrong.
The aircraft of all these companies are not UK registered- they belong to the respective companies.
Hope that helps,
Landing_24R