#1 is right search TT for Heathrow layover and see what you get..
However, it is a nice day in Dublin and I'm in a helpful mood.
Allow two hours on a good day from getting off the plane and getting to central London. Buy a Zone 6 travelcard at LHR tube station and get the tube. Go to Westminster station on the District Line, you can change at somewhere like Hammersmith to do this. This puts you in front of the Houses of Parliament ('Big Ben' or QE tower as it is really known as Fwoggie sez, used to be St Stephen's Tower), Westminster Abbey and Westminster Bridge all of which are great photo opportunities.
You are not really going to find a true 'English pub' in tourist central, they know their markets here and it is all a little bit or a real lot faux. However, The Red Lion in Whitehall (towards Trafalgar Square) might not be a bad choice. It is used by pollies and their minions so has some non-touristy aspect to it.
You have fish and chip choices at various places along Whitehall if it is lunchtime by then (including The Red Lion) such as Silver Cross (which does some great ales) right up near Traf Sq. Not cheap, not great food but these are options.
This puts you at Traf Square. More photo ops, feel free to dive into the National Gallery for a bit. You have, based on time, two choices I reckon. One, go up the Mall to Buckingham Palace and then across Green Park to Picadilly tube and back to LHR or two, go to Leicester Square Square and Picadilly Circus (and then again onto the Picadilly line) for more photo ops.
You are probably going to need to allow 2.5-3 hrs to get back from central London to LHR and checking in time and so on but that depends on the requirements of your airline, checked through baggage and so on.