With the first clue, we can determine that A2 must be a (P)atient. This is because it cannot be a doctor (since B2 tells us there are no doctors in column A) and it cannot be a nurse (otherwise, the A1 corner would be adjacent to 2 nurses, and it needs to be adjacent to one of each type).
We can also determine that B1 and B3 must both be (D)octors (to ensure that the A1 and A3 corners are both adjacent to one of each type) and that C2 must be a (P)atient (to ensure that the C1 and C3 corners are both adjacent to one of each type).
From the clues in A2 and B3, we can determine that C1 must be a (D)octor. This is because we need exactly 3 doctors in the answers (per clue A2), A1 and A3 cannot be doctors (per clue B2), and C3 isn’t a doctor (per clue B3). The only cell left for a doctor is C1.
From the clue in C1, we know that A1 and A3 must both be (N)urses (in order to have 3 Nurses in the quiz, per clue A2). This means C3 must be the third and final (P)atient.