The practical issues relate to income tax and social security payments on your husband’s salary. Where is his employer on these matters?
Income tax
Is the US employer obliged to pay tax under the UK PAYE system? See for instance this HMRC guidance
https://www.gov.uk/hmrc-internal-manuals/paye-manual/paye81610If the salary is paid via PAYE, this should take care of UK income tax liabilities.
If not, your husband should register for UK self-assessment, and pay tax directly to HMRC.
In both cases, your husband will need a National Insurance number. Has he applied for this? He may need to go in person to obtain this. If he needs to file a self-assessment, he will need also to apply for a UK Unique Tax Reference. A UTR is needed to enable a tax return to be filed.
Social Security
There are different rules relating to Social Security. There is a US UK social security agreement. See for instance the table on page 2 of the following guide.
https://www.ssa.gov/pubs/EN-05-10199.pdfOverall
Your husband’s employer should have taken professional advice on this, and shared it, before the secondment started.