Welcome! My recommendation is to apply for the Restricted license as well, might as well do it at the same time and cover all options.
Anyways, it is not illegal to reload in your house. However, there are certain home insurance companies and/or rental companies that have policies against "explosives" even though smokeless powder is not an explosive...
So, take a look at the policies and disclose information if necessary in order to ensure that you are insured. Some people have been screwed if there's a house fire and the insurance company discovers you had reloading components, even if the components were in no way involved with starting the fire the company might say they don't have to cover anything.
That said there are certain aspects of reloading that might be best kept to a garage or separate room, especially if you're going to be using a tumbler. There has been evidence that using a tumbler can spread lead residue around, so if you're doing it in your kitchen you might be getting lead dust all over your dishes. Not going to immediately kill you, but you might regret it in twenty years.
Casting especially, if you're going to do any casting keep it in a well ventilated area well away from anything else, especially food preparation areas and places where you might have splashing or spraying water, or loose papers and whatnot that might blow around and ignite. Exploding molten lead is no joke.
There are some regulations concerning the storage of primers and powder, I do believe the gist of it is to use a locking container, that can be removed from the house in the event of emergency, made of nonferrous materials labelled "Explosive" in contrasting colors so that the IRA know exactly where you keep the good stuff and can easily remove it from your house

. It's one of those feel good things that fall short of the intent, your canned foods are more of an explosive hazard, along with your aerosol cans, and synthetic fabrics, and all the other things that burn and explode just as well as smokeless powder. The real danger is using an airtight container that would allow pressure to build up to critical levels before rupturing, that's what you'd need to avoid.
A general rule of thumb is to wash your hands after reloading, you'd be surprised how dirty your hands can get. I myself avoid tumblers, I've got some large rubber gloves and some scrubbing pads and thrash around in the sink. Works quite well and it seems to contain the mess.
Happy reloading. Be sure to pick up a good reloading manual, or two, or three. Double check your numbers and you should do fine.