What is your understanding of what HH will do next? What have they told you about the rest of the process?

I agree in general but if the account is secure and cannot be accessed by anyone until the child turns 18, and it attracts a higher rate of interest, then there are definitely a few judges who will see the benefit. As always, it's a bit of a lottery!

Why not send the papers to both addresses?

That information sounds completely wrong. As the Claimants are minors the case should be settled by way of an infant approval hearing, essentially the judge reads the papers, sees the children and makes sure the settlements are fair for them. Assuming the settlement is approved the money is usually invested in the Court Funds Office until the children each turn 18, the fund applies interest at 0.5%, but all you would need to do is ask at the hearing for the money to be paid into a different fund. It's up to the judge but if the money will be going into a secure account attracting a higher rate of interest then it's a perfectly proper request. Simply making the request at the hearing should not cost any more than you would have to pay for the hearing anyway.

It's sadly not unusual for the order to omit any reference to the application for leave to appeal. I assume from what you've said that permission was refused but the judge told you that you had 14 days in which to submit a written application to the higher court for leave to appeal. You need to get the relevant appeal form from your local court and submit it. Assuming that your small claim hearing was before a District Judge in the county court your appeal will lie to a Circuit Judge in the same court, not the Court of Appeal as suggested.