How widespread is the usage of Senior, Junior, III in British English?

It is very rare in Britain for children to have exactly the same name as one of their parents.

Occasionally when there are multiple siblings in a single institution (i.e. a school), they might be designated 'major', 'minor', etc. to avoid confusion. However, it is an old-fashioned convention, with only public schools carrying on the tradition.

Outside of these institutions the designations are not used, mainly because children are usually referred to by their first names rather than their surnames.

(Note, in Britain the term "public school" refers, a little confusingly, to an older and more prestigious sub-category of private school. "State school" is broadly equivalent to what the US term "public school".)

Furthermore, in Britain, and in many if not all European nations, it could be regarded as a pretension to style oneself "so-and-so IV", even if only informally. Post-nominal numerals are generally the domain of royal and/or ancient houses and institutions (e.g. Queen Elizabeth II, 22nd Earl of Shrewsbury, Pope Benedict XVI, etc.).

The US never had an aristocracy and so the same consideration would not apply - or at least not to the same degree.

Historically firstborn boys were very often named after their father and firstborn girls after their mother. My family tree has four successive Williams in the 19th century, and the practice was widespread before then. 18th century parish records might show "John son of John Price jnr was born", or even "John son of John Price jnr. (cooper) died"; possibly there were three John Price jnr's.

I'd guess these usages were only on documents where the identification might matter at some point in the future; in normal speech you'd just say "John" or "tall John" or whatever your audience would recognize.

I believe English law doesn't have the concept of a "legal name". You can choose to be called whatever you like; authors may adopt several pen-names, entertainers have to adopt a unique stage-name. You can change your name by "deed poll", but I think that's merely a formal announcement of your decision. Should you appear in court, you might be asked "And are you normally known as Jack the Ripper?".