In John Ormsby's 1885 translation of Don Quixote, the word "despatch" is used. Is that the corresponding British spelling for "dispatch" or is it simply an archaic spelling (in both the American and UK English dialects), or is it both (British and archaic)?
The OED lists both spellings with equal status. 'Dispatch' is by far the more common spelling, uniquely so in the 16th, 17th, and 18th-century examples. 'Despatch' seems to have become fashionable in the late Victorian period.
When I was a child in 1950s' Britain I well-remember it often being spelled that way. But it has gone out of favour again. Nowadays you do still occasionally see 'despatch' used. Many would consider it old-fashioned but I am of the age where that becomes more a compliment than a reproach! It is the sort of thing I do, like referring to denims as 'dungarees' just to annoy my children.
Despatch is still quite common in the UK. The manufacturing company I work for has a Despatch Team for example. In my view, in the UK despatch is traditionally used in preference to dispatch but that appears to be changing with younger people, who are more likely to use the American spelling.