Well, in the north-west of England, where I used to live - North Staffs, Cheshire, North Shropshire - pubs serving food served pretty decent food. Not gourmet-book-in-advance sort, but definitely not microwaved either. They had a full kitchen and served from a full menu, where a lot of the food would have been prepared in advance but still came out piping hot and delicious.

Our local favourites don't seem to have websites, but here's a detailed TripAdvisor review for a pub five minutes' walk from where we used to live and the food from which we loved:

White Hart.

So if you're looking for quick 'n' cheap diner-type food, a café may be the best equivalent to a diner - but I certainly wouldn't equate a diner with a trattoria or bistro, just as I wouldn't with a pub! I consider diner food to be much lower in quality, mass-produced, frequently microwaved from frozen and very much short-order food. An old-style family restaurant in the US might be closer to the pub/bistro thing, but not a chain restaurant (Bob Evans, Cracker Barrel etc) - you just don't get many chain restaurants in Europe, and those that do exist are considered inferior - the Little Chef/Happy Eater type, for example, which is closer to a US diner or chain family restaurant.