I did indeed do it the end of May/beginning of June 
Obviously UK weather is a bit of a lottery, but had great cycling weather all through England, about 22 degrees with virtually no wind (got burnt to a crisp the very first day), and then it rained every single day in Scotland bar the final day and was fucking freezing (it was the first year they’d ever had the ski slope at Glenshee open on midsummer’s day). But yes, overall good time of year - pre school holidays so nowhere is particularly busy. Although don’t try and stay around Appelby when the horse festival is on like I did…