You are being very optimistic to assume that the taxi driver
will waive the fare. Remember, this is SINGAPORE after all, where
everything is about money (even our ministers need to be well-paid
to be corruption free).
If it has been a genuine case, I don't see anything wrong with the
mother borrowing the money to pay for taxi fare and the hospital
registration fee. After all, it only take like at most 1, 2 minutes
to borrow the money rather than call the ambulance, wasting time
giving them direction to come, and wait for the ambulance to be
dispatched.
Yes, if the victim is wounded, it is probably dangerous to move
him. However, in this case, it's a baby with severe fever. The best
thing to do here is to grab a cab and go straight to the hospital
rather than wait for the ambulance to come.