When is a good time to toilet train my cats?

I have finally my limit with cat litter. It’s suddenly EVERYwhere, not just on the floor by the box, but tracked into the hall, in my rocking chair, on my dining room table, etc. I’ve always swept, but this is nuts. I want to train my 2 cats to use the toilet. I have a half-bath that is already exclusively theirs, so I won’t be losing out on any personal convenience (won’t have to move a tray every time I need to use the bathroom).
The dilemma I’m facing is timing. I have a baby due in a week and I know that is going to be stressful for them. One of them also might need surgery that I’m going to need to time with the baby. I’m not cracked enough to re-train them while one is recovering from surgery, but should I try to get it done now or wait until afterward any recovery period I might be facing with my oldest? I could also be moving in as little as 9 months. Should I just suck it up and deal until we move? Any thoughts, except the mean ones, are appreciated.
To clarify, they ARE already litter trained, I DO have a covered box, and outside isn’t an option, we live in a 3rd floor apartment with a secured front door.
Toilet training a cat wouldn’t be as intensive as house-training a puppy, only a matter of switching the training trays and keeping the litter in them clean (which I already have to do), but I DO realize I will have a baby to look after.
I also appreciate your concers about the toxoplasmosis. I assure you that I know about it, I’m clean, the cats are clean, and if I had a choice (hubby’s deployed, sorry), I wouldn’t be anywhere near the box.

Is it a good idea to toilet train a cat?

I bought a kit for toilet training my cat but I’m nervous about using it since it requires removing his litterbox. I welcome any encouragemnt, ideas or suggestions!