Is it possible to have the computer analysis report mate in 12 moves to only change that to mate in higher number of moves, like 15 for example, after it searches deeper?
It is possible, if the computer program has a bug... A position is assigned a win-in-(k+1) only when one move from that leads to a loss-in-k, and assigned a loss-in-k only when every move from that leads to a win-in-k or when k=0 and it is a checkmate. So every correct chess program will never assign a position to win-in-12 and then later revise it to win-in-15 due to "searching deeper". No heuristic pruning techniques should ever affect this, as heuristic pruning can only weaken the evaluation away from a win/loss (in a correct program).
However, it is possible that the program is designed to search deeper if the input position satisfies a certain criterion C, in which case it may find a win-in-12 for a given position X that satisfies C, but then one move later the resulting position Y no longer satisfies C and so the program does not search as deep lines as it did for X, and so fails to find a win-in-11 for Y! It may still be able to prove a win-in-15 for Y if it has some rudimentary endgame evaluation. For example, KQK is always a win-in-10, so we do not have to do any search to return win-in-10 in such a position. Thus the program might do a more shallow search for Y and see a KQK in the search tree and immediately return win-in-10, but a deeper search would have revealed that it was actually win-in-5. This can produce the observed effect, but is due to "searching less deep"!
Note that clicking on "go deeper" on Lichess analysis may not use the same memoization table that had been used before at the lower depth level. One clear example is here where the cached analysis states #−30 but if you click "go deeper" it clearly starts completely from scratch. This is expected; after all it makes no sense to store huge tables along with the cached analysis. I think that if you cannot provide an actual concrete example of the behaviour you are asking about, it is difficult to address your hypothetical question more than I already have.