If a pawn is promoted to queen adjacent to the enemy king, is the king instantly in check?
Per FIDE Chess Laws, rule 3.1, a piece is said to attack an opponent’s piece if the player’s piece can make a capture on that square (and in one more special case not relevant here). In full:
It is not permitted to move a piece to a square occupied by a piece of the same colour. If a piece moves to a square occupied by an opponent’s piece the latter is captured and removed from the chessboard as part of the same move. A piece is said to attack an opponent’s piece if the piece could make a capture on that square according to Articles 3.2 to 3.8.
A piece is considered to attack a square even if this piece is constrained from moving to that square because it would then leave or place the king of its own colour under attack.
Per rule 3.9, the king is in check when it is attacked by one or more of the opponent’s pieces.
Therefore, as soon as the move is completed, the king is in check.
If the pawn is adjacent to king before promoting, it's already in check. For example, pawn is on e7 and King is on d8, king is in check.
However, if pawn is 1 rank away from king and is promoted to queen, yes king is instantly in check. For example, pawn in on c7 and king is on e8, if c8Q, king is in check.
It is applicable for all the pieces not only for Queen.
I initially thought that it isn't possible to promote a pawn on a back rank square that is adjacent to the king, as that would suggest that the opposing king was left in check on the previous turn. But remellion pointed out that a pawn could promote on a square adjacent to the king if it captures on its promoting move.
Either way, assuming no pieces are between the promoted pawn to queen and the opposing king, the white king would be in check on this next turn.