https://www.british-history.ac.uk/survey-london/vol27/pp21-23
St Mary Spital started in the 13th century just outside of London’s Bishopgate. Later it was called Bedlam. DuckDuckGo has many articles about it.
Could this be an earlier hospital attribution?
You are probably right. In Germany the same word, “Spital” is regarded as dated but it still survives in Austria and in Switzerland.
Here I found a vague reference to a cognate, “hospitālhūs” (“hospitable house”?) dated to the 11th century, while for “Spital” as such the same source doesn’t provide any dating prior to 1894.
However in all cases the etymology seems to be the same, from Latin hospitalis - hospitale (in the sense of “shelter for people in need”).
Thus it’s a legitimate suspicion that the same word, more or less locally mispronounced but substantially intact, has been traveling along the whole Europe for much longer than the 10 centuries we may assume.