I asked exactly the same question last year, prior to our visit at the end of May.
You can find some excellent advice I was given in this thread:
Farne Island & Northumbria - Advice Please.
We went around the last week in May and had a great time, unfortunately we only managed to land on the islands once, due to bad weather, and that was at the end of our trip.
We may well go again this year but it will be around the second week in July if we do, we would hope that the seas will be calmer and the weather warmer at that time.
I would thoroughly recommend going though, even if you can't get on the islands there's lots to see and some beautiful countryside to walk in and around the area.
Dave