Six days for the "big" semi-loop seems reasonable, with stops (roughly) at Little Yosemite (short/steep first day), Lower Sunrise Lake, McGee Lake-Glen Aulin, Waterwheel Falls-Return Creek, and Pate Valley.
The switchbacks north of the Sunrise Lakes junction stay snow covered until fairly late, and crossing Tenaya Creek when it's high can be a challenge (there's no bridge). Last weekend I was at Lake Vernon (north of Hetch Hetchy) and there was still significant snow on the south side of the Grand Canyon of the Tuolumne, so I'd also expect snow between Pate Valley and White Wolf.
Note that even if the Tioga Road opens, the spur to White Wolf may not open at the same time, so you may need to hike along the closed road a few miles, but you'll want to start early anyway to avoid the heat while climbing out of Pate Valley.
For the shorter loop, you may also encounter significant snow between Snow Creek and North Dome.
Either version should be wonderful. Have a great trip!