As This Person Who Writes Stuff just mentioned, you have the actual Warner Bros studio tours of Harry Potter sets, costumes, props and such, as well as tours offered by various companies for filming locations in and around London.
I've yet to actually do the Warner Bros studio tour, but did go on a half-day Harry Potter + parts of London tour with Capital Taxi Tours for HP filming sites and locations (this was before the Warner Bros studio tour even opened). That was a tour with a Blue Badge guide and black taxi driver that was amazing - so good, in fact, that we canceled a group tour we had scheduled for a couple days later to book a new full day tour with Capital Taxi Tours (of Hampton Court Palace and Windsor). It wasn't cheap, but worth it for us, and if you check out the link you'll get a good idea of what sites are included on their tour and - even if you don't plan to book a tour with them - what filming locations are in and around the city as opposed to in the Warner Bros studio tour. Kings Cross, Grimmauld Place, Gringotts, The Leaky Cauldron - these are all filming locations you'll see on a regular Harry Potter themed tour around town. Keep in mind, though, that oftentimes visiting these filming locations means simply seeing them from the outside. For security reasons you can't go in to the location used for Gringotts, for instance, because it's the Australian High Commission (an embassy). Other spots on a tour might just be where exterior shots were taken - your imagination and memories of the movies will have to fill in the rest. They even offer a tour that involves seeing the London sites, then dropping you off at the Warner Bros studio for your pre-booked studio tour scheduling and booking that must be handled on your own, then returns you to your hotel. Again, their tours are on the pricey side because it's a private tour with your own driver/guide. There are other companies that offer more affordable group tours.
As you've seen, studio tours book out months in advance. It's late April now, and regular adult tickets are sold out until July. There's a Deluxe studio tour option that has much earlier availability (as soon as May), but those tickets are several times more expensive than the regular ones.