If you are planning to see Yosemite, then head south to see Sequoia, then your natural choice is to keep going south and cross over Tehachapi Pass on your way east. However, this will put you into some low desert on the way to the Grand Canyon, and it usually has begun heating up by this time of year. If you are acclimated to the dry heat, it's actually not too bad if you avoid the midday sun, but I don't know how you will do coming straight from England--are you a heat lover? (The last couple months here in Southern California has made riding in 100 F seem pretty normal..) The worst months in the low desert are in July-September when a "monsoon" flow kicks in from the southeast, bring lots of humidity, thunderstorms, and nights that barely cool down at all.
Oh, yeah, Death Valley just reclaimed the title of hottest place on earth at 134 F; the problematic record-holder from Libya finally was disallowed after umpteen years. That should make it more of a destination--just this summer it set a record for its hottest night ever during one of the monsoon high pressure episodes--I think it did not fall below 107 F.
PS: here's some opening dates for Tioga Pass in recent years:
http://www.nps.gov/yose/planyourvisit/tiogaopen.htm
The North and South rims of the Grand Canyon are at 8000 and 7000 feet respectively, and certainly will not be suffering any low desert type of heat, regardless if it's June or not. You may freeze your tukis off, instead...