Another vote for 3 prong Fanny pack! Only need the lockers for Hulk and Ripride rocket that way. I found a semi stylish one on Amazon that I could live with and it was extremely convenient to not have to deal with the lockers most times.
I did Universal Orlando Resort with a small crossbody bag a few years ago. I wasn't doing many of the rides where you needed to put your bag in a locker. On the few rides where I did need to put my bag in a locker it wasn't a big deal.
There are very few rides you would not be allowed to take your bag along. Pack your bag with what you normally take into a theme park and do no let the possibility of needing to store it in a locker change what you would like to have in the park with you on a day to day basis.
With a wand, you'll need a locker at:
IOA: Hulk, Hagrid's, FJ. Additionally, you may want a locker for any of the water rides (these aren't free)
US: Rip Ride, Mummy, Gringott's, MIB
I just purchased the vera bradley slytherin 3 prong fanny pack/cross body (mini-sling); you can wear both ways. Will they allow me to wear this onto the allowed rides? I don't ride the Hulk or RRR.
I don't know if they have package pick-up right now, but in the past, people were able to take the wand out of the box and have the box and bag sent to the front of the park to be picked up later. Personally, I would do Ollivanders later in the day or right before a mid-afternoon hotel break. On our trips, we usually do a "wand morning" when we carry our wands, do some experiences on both sides, and go relatively light on the rides. I also purchased wand holders on Etsy to make things easier. I know that A LOT of wands get lost because it is so easy to set them down and forget them.
With a wand, you'll need a locker at:
IOA: Hulk, Hagrid's, FJ. Additionally, you may want a locker for any of the water rides (these aren't free)
US: Rip Ride, Mummy, Gringott's, MIB