The credit is allowed only in the year the adoption becomes final.  (A special rule may apply if it takes more than two calendar years.)

If not used that year, it can be carried forward. I'm not sure whether credit on a late-filed return is still eligible for carryover.  Also, at one time the credit had to be carried back before being carried forward. I don't recall the year that change was made.

The credit for special-needs adoption this year is more than $14,000, regardless of whether anything was paid or how much.  They missed out on more than $40,000, back in 2015?  Who was doing their returns then?  No questions asked about adding three new dependents?