I've done pregnancies and births in two different hospitals and they had different policies towards children. One was zero children in any appointment ever. The other, they would allow older children in scans, for example, or younger children to come if looked after by another adult. The policy seriously will be the policy, though, and you should not expect to turn up with your children and be seen if they've said you can't be.
WRT them not changing the appointment, if it's some bloods that can only be done at a certain stage of the pregnancy then you may miss out on this test. If they don't have an appointment to offer, they don't have an appointment to offer. On the other hand, you may be able to get a cancellation or similar.
I do honestly understand your husband not being able to take time off work, but none of the options available to you include you taking your children with you if they've said no, I'm afraid. You can either beg and grovel for an appointment change; or you can decide not to have these blood tests and deal with any consequences; or you can find a babysitter.
It's pants. It really is. In the first hospital I was at, an appointment would take fully half a day with all the hanging around waiting for them and I'm so glad I didn't have a child already as childcare would have been a nightmare. With the second, I could actually expect to be seen at my appointment time, which made things so much easier.