My child is entitled to US citizenship but we have been unable to register her birth abroad as due to covid we cannot get an embassy appointment. She has a valid UK passport. We need to travel to the US, will she be able to enter the US on her UK passport? She would be travelling with me (UK passport holder), my husband (US passport holder) and our other child (US passport holder).
Your best shot is to try for an ESTA.
The ESTA application will ask you specifically if you have other citizenships and you need to answer truthfully. The rules about ESTA for US citizens are unclear and there have been cases where an ESTA was issued without problems. As with many things CBP, the rules are vague and there is an unfortunate amount of "officer discretion".
If there is a field for comments in the application, add that you couldn't get a passport due to Covid. There are currently Covid exceptions for expired passports so maybe their is one for non-issued passports as well.
If that gets denied, you can consider a land border through Mexico or Canada. These are currently restricted due to Covid, so you need to read up on the exact details of the current conditions. The current restrictions are in place until June 21, but what happens after this is anyone's guess. See https://www.reuters.com/world/us/us-land-border-restrictions-extended-with-canada-mexico-2021-05-20/#:~:text=United%20States%20land%20borders%20with,extended%20in%2030%2Dday%20increments.