I don't think that transfers in January have ever been of such a significant importance that they helped a team win the Champions League. If that was the case, I wonder what type of player you are talking about. Of course, if Real Madrid would sign Mbappe in January and he would score several goals in the final round of the Champions League, then it would perhaps be true, but since that won't happen, I doubt that any transfer in January would literally be responsible for the Champions League trophy at the end of the season.
I already said what type of player I was talking about. A big transfer like Mbappe could change the direction of the Champions League for the team transferred him. We can't be 100% sure about this transfer not to happen in January at the same time. I just feel like it would happen this summer probably rather than January.
But it is really difficult to be sure of something in the transfer market. It is all about money and what that player feels about this transfer to happen immediately.
Since Mbappe is available for a contract offer we never know.