HEROES Act Passes
I would start with describing the HEROES Act as an opening bid in a negotiation with the Senate, where Republicans and most Democrats are substantially more cautious. It is worth noting that this summary does not include the pro-bankruptcy provisions, including the $100,000 homestead minimum regardless of state law, the more general allowance of 84 months plans (but with a discharge still at 36 or 60 months), etc.
Matt is correct that, absent the bankruptcy provision, the federal government cannot cancel rents, mortgage or other debts without being an unconstitutional taking. Even under the expansive exercise of its war powers, the federal government cannot take property. without compensation Bankruptcy is the only exception I know of.
Under the 10th Amendment and the full faith and credit clause, the federal government can't suspend garnishments that are valid under state law. Perhaps it could, under the Interstate Commerce clause, interfere with out-of-state garnishments, but even that would be problematic and rather rare. Again, as a North Carolinian that gets clients without any real wage garnishment, I can only look longing at those of you that have it.
Since garnishment requires the exercise of a State's police power, it could hold those off for emergencies. I suppose that a State could order that all repossessions are a "breach of the peace" making them illegal under the UCC.