The key thing regarding expenses and HSA is the date the service is provided and the date of the "split" of responsibilities. On the date of service it doesn't matter if they have their own policy, or the type of policy they have.
I am going to assume that the HSA already existed at the time that service was provided, because if it was established after the date of service the question is moot.
The HSA funds can be used by the tax household, which is the married couple and the tax dependents. If the date the medical service was provided was before the date that the court severed responsibility, then the HSA funds can be used without tax consequence.
If the service was provided after that date, then if HSA funds are used there are tax issues.
The bill can be paid days, weeks, or months after the date of service. We al have experienced long delays sometimes before the final bill is determined.
I am basing this on the instructions for form 8889 line 15:
Only include on line 15 distributions from your HSA that were used to
pay you for qualified medical expenses (see Qualified Medical
Expenses, earlier) not reimbursed by insurance or other coverage and
that you incurred after the HSA was established. Do not include the
distribution of an excess contribution taken out after the due date,
including extensions, of your return even if used for qualified
medical expenses.
In general, include on line 15 distributions from all HSAs in 2019
that were used for the qualified medical expenses (see Qualified
Medical Expenses, earlier) of:
You and your spouse.
All your dependents.
Any person who would be a dependent except that:
The person filed a joint return.
The person had gross income.
You, or your spouse if filing jointly, are dependents of someone else.
On the date of service they were still your spouse.
