During the chess game, black (Jay) captures the white queen with his king, after which white delivers check with her bishop. When black moves his king out of this check, the white player asks 'You don't want to castle?'. You can't castle with a king that's (a) moved and (b) is in check. A chess grandmaster would know that.
Henry is shown going to bed in the US while talking to Elizabeth on the phone. Elizabeth is in Ireland and it's daylight. If it was night time in east coast America it would be late night in Ireland and dark outside.