It could be that they had first met earlier in the day, perhaps for tea with the Princess of Asturias, and so it did not require a bow. Although one might consider lowering ones head ever so slightly to kiss the hand a bow. The PoW is pretty good about observing protocol.
The Queen looks lovely and like she is enjoying herself. I love these kind of engagements. Her hair is also looking extra lush, I like this look on her.
I doubt they met for tea with the Princess of Asturias since she is still at school in Wales, which is quite far from Bishop Auckland (in northern England).
The Prince of Wales seems to be very close to the Spanish royals, which is not surprising since King Felipe VI is his second cousin once removed in the Greek line and his third cousin once removed in the Battenberg line.
Greek line
George I of Greece -> Prince Andrew of Greece and Denmark -> Prince Philip, Duke of Edinburgh ->
Prince Charles, Prince of Wales.
George I of Greece -> Constantine I of Greece --> Paul of Greece --> Sofía, Queen Consort of Spain --->
Felipe VI of Spain.
Battenberg line
Prince Alexander of Hesse and by Rhine ---> Louis Mountbatten, Marquess of Milford Haven---> Alice, Princess Andrew of Greece ---> Prince Philip, Duke of Edinburgh -->
Prince Charles, Prince of Wales.
Prince Alexander of Hesse and By Rhine ---> Prince Henry of Battenberg ---> Victoria Eugenie, Queen Consort of Spain ---> Infantte Juan, Count of Barcelona --> Juan Carlos I of Spain --->
Felipe VI of Spain
In the Victoria line, I believe they are fourth cousins.
Victoria of the United Kingdom--> Edward VII of the United Kingdom --> George V of the United Kingdom --> George VI of the United Kingdom ---> Elizabeth II of the United Kingdom --->
Prince Charles, Prince of Wales.
Victoria of the United Kingdom --> Princess Beatrice of the United Kingdom ---> Victoria Eugenie, Queen Consort of Spain --> Infante Juan, Count of Barcelona --> Juan Carlos I of Spain --->
Felipe VI of Spain.
An alternative Victoria line for Prince Charles (via Prince Philip) would be:
Victoria of the United Kingdom ---> Alice, Grand Duchess Consort of Hesse and by Rhine --> Princess Victoria of Hesse and by Rhine --> Alice, Princess Andrew of Greece ---> Prince Philip, Duke of Edinburgh --->
Prince Charles, Prince of Wales.