Nothing will be double taxed, but your husband's Roth conversion in 2020 will be largely taxed, with only a small amount of his basis being applied to make that portion of his Roth conversion nontaxable.
If I understand correctly, he'll have roughly has $280,000 in traditional IRAs at the end of 2020, has $15,000 of basis in traditional IRA contributions and did a Roth conversion of $19,000. If these amounts were exact, this means that the nontaxable portion of the $19,000 Roth conversion would be:
$19,000 * $15,000 / ($280,000 + $19,000) = $953
and the remainder of the conversion, $18,047 would be taxable. The $14,047 of basis that is not applied to this conversion will remain with his traditional IRAs to be applied to all future distributions from his traditional IRAs until he has no more money in traditional IRAs at the end of the year that he finally drains all of his traditional IRAs. (This is why there is no double taxation.) The calculation is done on Form 8606 (or partially on TurboTax's IRA Deduction Worksheet if TurboTax shows asterisks on lines 13 and 15 of Form 8606.
If your husband can roll the $280,000 to his 401(k) with this distribution from the IRA occurring before then end of 2020, that would leave a $0 balance in traditional IRAs at the end of 2020 and would allow all $15,000 of his basis to be applied to the $19,000 Roth conversion. The rollover to the 401(k) can only be from pre-tax money, so none of the basis moves to the 401(k).
