The islands were ruled as a protectorate by the British having taken them off the French from 1810 until the liberation of Corfu in 1814.
The islands were ceded to Greece in 1864 as part of the campaign to bolster the prestige of George I, King of the Hellenes. So, the POD would seem to be that George I doesn't become the Greek King. One way would be to have Prince Albert live and he might have persuaded Victoria that Alfred, Duke of Edinburgh, would be a good choice for becoming a Greek King. There was also the not inconsiderable fact that Prince Alfred was very popular with the Greeks having won 95% of the vote in a plebiscite to choose the next monarch.
Alfred therefore becomes King of Greece and the British, as part of the quid pro quo, retain the Ionian Islands as a British protectorate.
What then ? How does a British military and naval presence in the Adriatic affect developments regarding Albania, Italy and the two World Wars ? I've some ideas but over to you all..