Hi Op
So sorry you are in this difficult position. I remember trying to register ds3 in town with no parking spaces for miles, I had bronchitis, it was new year - my Dad was luckily able to drop me there with him, I could hardly stand up but managed to stagger into the building and up the stairs and it did take at least 10 minutes to do all the forms.
I then staggered out again and my Dad came back for me. My Mum was meanwhile managing my older two. (I was on my own as well - sometimes there is no one, I'm so lucky to have my parents nearby).
I don't think I'd have coped without my folks and I totally sympathise. You are doing well to cope as it is. I think you sound sensible and caring and you're doing your best in a very tricky situation.
I would first off ring the registry office again and explain the situation. They MAY have a cancellation in the next few days or next week if you're lucky, then you can relax. They will be aware of the appointment situation already, it's not your fault.
I would hesitate to take your poorly boy out tomorrow, buggy or not, as he will feel really rotten. I wonder if, it being late already, they will do their utmost to find you another appointment.
Worth a try when they open? xx