Hello,
June shouldn't be too bad - not school holidays or anything.
I'd do Scafell Pike and Helvellyn on weekdays if I was you and maybe more obscure ones on weekends (Helvellyn in particular can get very busy on summer weekends in my experience).
You could always try different routes - a good one for Helvellyn is from Patterdale, head up Grisedale valley and tackle the ridge up Nethermost Pike, then back down Striding Edge, hopefully the traffic heading up the edge the other way will have calmed down by then. Scafell Pike from Eskdale is a good idea too, although transport will be more difficult - can't really advise on buses, I suppose it'll be on the web somewhere....?