When you talk about the "large nut," you don't mean the one that the valve is poking out of, do you? Don't loosen that.
Turn the water off - valve handle is right there.
Loosen the threaded hose end with an adjustable wrench or medium-large pliers. It shouldn't take much effort to do since that water supply is equipped with a rubber washer inside so it's easy to make a good seal. It's just like a garden hose, except smaller. The hose will hold still while the threaded end turns.
Thread on an add-a-line T fitting. Any plumbing section should have this and any competent associate at one of the big box stores or, even better, Ace Hardware will be able to point you toward the right kind of fitting. The one you want has to have a 3/8" compression FEMALE, and two 3/8" compression MALE ends.
Look at the fitting ends of the hoses. The one going to the toilet is the right kind already; your other line going to the diaper spray, peek into the end that threads onto the new T. If there's a washer in there, you won't need to bother with Teflon tape. The washer will do well enough to snug everything down and prevent leaks.