I worked at the Tuolumne Campground for three summers, and this is my take on camping there.
As is commonly mentioned, the campground is half reservation and half first-come first-served. The reservation slots go on sale several months prior through the phone number and internet site that are already posted. The first-come first-served sites vary in the numbers that are available everyday. The campground is large (300 sites), but the total number of sites sold on a first-come first-served basis can be as little as 10 per day, due to campers extending their stays once they arrive.
Between July 4 and the end of August there is almost always a line at the campground office when it opens. It was my experience that most of the people who were there at opening time got a site, though this was not absolutely always the case (ie: don't blame me if you don't get one). Once the sites are officially sold out, the campground staff starts a waiting list for any openings that might arise due to cancellations or no-shows. There are usually a few of these, so it is worth putting your name on the list. Once the extra campsites are determined, the staff will sell the waiting list, and you MUST be there to get a site. If you need to run down to the Grill for some coffee, leave someone behind. They won't hold your site for you just because you were first on the list. That being said, they are all really nice people.
The staff will assign you a site at a kiosk inside the campground. Please be friendly to them and understand that juggling rservations in a 300-site camground is a really complex and frustrating job. You can put in requests for sites you like, and they will probably fill them if at all possible. A lot of folks like the "A" loop, because it is next to the river. I prefer the "F" and "G" loops because they are more remote and there aren't any RVs back there.
What is the closest wifi access for the meadows?
My guess would be Lee Vining. I think there is a coffee house in town (with delicious homemade muffins, btw) that has wi-fi access, but perhaps someone could suggest something closer?
You might try asking on the CA forum.