I've not had to do the new MOT after a cat D before but I've not done recommision CAT D salavge repair in a year.
Perhaps it's a coincidence the last cat D's I've had the MOTs had run out anyway.
But my Cayenne One was a still valid (last year) and I taxed it without getting a new MOT maybe it's a new thing or certain insurance firms do this.
Safe bet is to ask the firm that paid you out
Sorry I may have unintentionally got it wrong