I have a pizza stone, which makes a big difference. But I don't always use it because I have a pizza pan that's bigger than the stone. There are a few things you can try:
Roll the base thinner, so it cooks through quicker. When rolling I tend to flour quite heavily, which reduces the amount the sauce soaks in.
Cook directly on the oven shelf, with a baking tray underneath to catch anything that falls off. This seems to work better for ready-made/par-baked bases. I've never tried parbaking a homemade base but think it would be tricky to get the texture right.
If you like a lot of sauce, heat it just before topping the pizza. This will help the temperature of the base come up quicker.
Don't mess around putting the pizza in -- I've seen people open the oven, walk across the kitchen to fetch the food, position it just perfectly, change their mind about the shelf height, then get round to closing the door. Instead, put the shelf near the top before you preheat, and have the pizza in one hand while you open the door with the other, or at least within reach.
A good preheat is important for something that cooks so quickly, even if your oven doesn't boost the preheat with the grill (broiler) element. I turn the oven on before rolling out/stretching, so it's been preheated for some time when I put the topped pizza in (especially as my small person likes to help withe the topping so it takes a while). This means that all the metal in the oven is up to temperature when it goes in. Note that I don't preheat the pan, as I stretch the base to fill it, then top the pizza on the base -- it's a bit unwieldy otherwise.
Cooking two pizzas on two shelves makes things much harder -- that's partly why I use my (rectangular) tray instead of my stone, and make one big pizza.