Personally I’d say May or September. They tend to have better weather. And you avoid the midges of midsummer. But there have been remarkably few midges this year and June was lovely, so you could hit a good year - or a bad one. Keep between the Solstices so it’s lighter later: sunset is about 9pm at the moment.
I completed the Way last in the last week of last April and that was pretty good. Just close enough to summer to be warm and (on all but one day) dry, just far enough to be free of midgies!
I think the kinda beginning or end of summer is generally a good time.
Amazing family weekend with old steam engines, classic car displays, market stalls, and full catering and bar. And camping on site - Save £25 by booking in advance.