Where do I stand re working the day? If you really have absolutely no-one you trust to take care of your child on that day then you have no choice but to take the day off, could another vet nurse swap with you? I guess you have tried that already though.
How old is your child? Would your place of work (veterinary surgery I take it) be insured to allow your child to sit in the office with a colouring book, some games or something to occupy him/her, I know this depends on age if very young obviously the child could not be left unattended.
If you explain that there really is no-one to take care of your child on that day, then it will be up to your employer to use that day as unpaid leave, a holiday or to sack you (rather stupid after all your training to do that though) depending on the time you have been there and just how much notice you have given for this change in circumstances, your employer does not have to give you that day off if it is really inconvienient or even impossible for them to work around, in anything else it is down to you to make sure you have cover and to work the days you are down for, but as this is a child obviously none of us are going to say leave your child, you have to work!

I've never let anyone look after my child I don't know, so feel a little hypercritical in suggesting such a thing, (infact a lot) but are there any nannies/childminders you know of through friends who you could try to book for the day? Just another thought, otherwise you will just have to ride out the day and hope your boss does after all have a heart, you have no other choice, your child comes first.
Good luck!