Once you cross economic thresholds in different states, the UK LTD will need to obtain an EIN (now you can call or fax, so you have to mail it) and get registered for sales tax in those states. You will need to collect and remit sales taxes likely no a quarterly basis.
The challenge will be without a US bank account, and you will not be able to use the automated software to file your taxes. They need to do an ACH pull from a US bank account. Your options are to form a US LLC with a bank account owned by your UK LTD (which we can help you with that process see https://www.nvinc.com/us-entity-formation-launch-with-confidence/) or work with a tax firm where you may wire sales taxes to them to file your returns on your behalf. Yes, some states do allow you to pay sales taxes with a foreign credit card, but most do not, so it is cumbersome.
Here is a chart and video that may be helpful: https://salestaxsystem.s3.amazonaws.com/Sales_Tax_Compliance+_Economic_Nexus_Shopify.pdf Separately, if you are dropping that will involved exemption certificates also.
I hope this helps.