I was just on that trail July 10 and 11 so can give you the scoop on conditions as of right now, but the melt is on and conditions will have changed (improved) considerably by the time you and your group get there.
Trail conditions:
Tuolomne Meadows to Glen Aulin was snow free all the way. The stream crossings at Delaney and Dingley are shin to knee high, the water was cold but not really moving much, so I got my feet and legs wet and cold but it's not dangerous. I also got wet crossing the bridges. The Tuolomne river was running so high that the area next to the bridges were flooded, again it didn't feel dangerous, just wet.
Glen Aulin to May Lake was tougher. Snow line starts at about 8400 feet in among the trees which is about two or three miles from Glen Aulin. It's wet sloppy snow with lots of post holing among the boulders and the trees. Map and compass skills needed. Cathedral creek was about like Delaney, knee deep, wide and cold. At the May Lake / Murphy Creek junction I decided I preferred to check out Poly Dome lakes so went down the Murphy Creek trail and can't comment about the rest of the way to May Lake. It was wet melty sloppy snow the rest of the way to the larger Polly Dome lake, but easy to find a snow free campsite up on the rocks overlooking the lake. Had it all to myself and loved it.
The next morning I hiked out on the Murphy Creek trail which had some snow almost all the way down to Tenaya Lake at 8100 feet.
Backpackers camps:
Tuolomne Meadow car campground and backpackers camp is due to open this Friday July 15th. You can stay at the backpackers camp with your group one night before and after your trip if you have a permit. Fee is $5 per night per person. The backpackers campground has about 20 sites and needs to fit all who arrive, so grab a site at the edge and put your whole group in one site. Each site has a bear box, picnic table and fire pit. The bathrooms have flush toilets and water spigots, but no showers and are located in the surrounding car campground.
Glen Aulin backpackers campground is adjacent to the High Sierra Camp. It is free to stay here.There is a composting toilet for the backpackers. There are also a few bear boxes, but not one for each site. I like the upper sites better that are farther away from the HSC, they are more spread apart, have nice views, but are further away from the bear boxes and toilet.
May Lake backpackers campground is adjacent to the High Sierra Camp. It is free to stay here. There is one flush toilet and one (or maybe two) bear box for all backpackers to share. There are not specific sites, just a general area for backpackers to set up there tents. Camping here next to the lake is beautiful.
Hope that is helpful.
