I'm lame, and I don't know if you literally mean a carriage, like a baby stroller or a carriage as in a vehicle, but the following advice applies to either:
So, you're going to need to go heavy duty here. If it were me, I would pull the carriage out so it can get some fresh air. If we're talking about a vehicle, open the windows too.
Next, if there are any removable upholstered parts pull those out and throw them in the washing machine on hot with a huge scoop of OxiClean. I'd do an extra rinse before throwing it in the dryer.
For anything else that is cloth/upholstered you're going to need to get a portable carpet and rug machine.
Spray everything down with an OxiClean spray and let that sit overnight. Then, use the machine and an OxiClean based formula to soak, scrub and wet-vac up all of the stains and affected areas. I've cleaned my own car this way just to freshen it up after it mildewed a bit inside from our snowy boots.
Once you've gotten the carriage washed, keep it outside to air out and dry properly, so it doesn't produce mildew.
I suggested OxiClean products a lot because that's what I use on almost everything. Don't use bleach, even though it's tempting. If the cats urinated in there at all, then there's ammonia, and you don't mix ammonia and bleach otherwise you get a dangerous chemical reaction.
If you can't get OxiClean based products let me know what country/region you're in and I'll try to find an alternative that's available in your area.