The Daily Programme for the QM 2 (I think also true for QV and QE) states "Tonight's Dress Code: xxx) which applies thruout all the dining venues on the ship -- Queen's Grill, Princess, and Britannia. The code goes from (a) Formal - Tuxedo/Business Suit & Tie, (b) Semi-Formal - Suit/Sport Coat & Tie, to (c) Elegant Casual - For Gentlemen: A jacket, no tie required. Ladies wear, consistent with those for gents.
On a 7 day transatlantic crossing, about 3 days were Formal, and the others split between semi-formal and casual.
For Queen's Grill, I would guess the dress would be more formal than what the code might require.