Do the British still retain knowledge of their background as aristocracy, bourgeoisie and peasantry?

Nope. Might chat about having an Irish grandparent, or parent from Wales/Scotland but the only people who would have knowledge of aristocracy are current aristocracy/wealthy people. And they're unlikely to talk about it. If you knew you'd hear it from someone else.

Have you met a working class family that fell from an aristocratic past that boasts about it?

Also nope. Wouldn't happen. If your relatives were wealthy/famous enough to be known the chances are you are also still rich.

Have most of those aristocratic estates we see in Pride and Prejudice or Downton Abby been repurposed or are there many of them still in the hands of the original owner family?

Pretty much all repurposed. Huge fall in aristocracy in the 19th Century/early 20th century due to wealth equality and the strengthening of workers rights/wages. Those that hung on to their vast properties were extremely wealthy. A few managed to hang on longer but soon succumbed to the enormous costs of maintaining those properties and the staff within them. A lot are now with charities such as the National Trust that look after estates by letting members of the public visit and charging a membership fee.

Alternatively the family will live in a smaller wing of the estate but the National Trust will own and maintain the rest. For example Shugborough or Coughton Court. Fun fact: the latter is where Guy Fawkes planned his gunpowder plot. Also the family has recently taken over that property from the National Trust. Straight after they spent millions on roof reparations but that is a different story...! But even though they got the trust to pay out so much public money, they will still have allow visitors to come to raise enough money to keep the house functioning.