we will have a 4 hour layover in London on our way to Budapest. Is it feasible to see anything or just take a taxi ride for a couple of hours?
No. It would take you at least two hours to get through customs and get to the city, most likely.
There's no way you can get through customs to get out of the airport and and then back in through security in 4 hours much less go all the way into London from the airport...London traffic is terrible so a taxi would take you a good 30 - 40 minutes just to get into town. By tube it's about 45 minutes to central London one way.
Nancy, which of the 6 London airports are you arriving into and from? Is it the same airport you are leaving for Budapest from? I don't know how you are calculating your 4 hours. If you are in and out of the same terminal at Heathrow, and if you have calculated the start of your 4 hours as starting after you have landed, gone through Immigration, baggage reclaim, and Customs, maybe checking your luggage through to Budapest; and counting the end of the 4 hours with 2 hours to allow you to check in for your next leg and clear security - - in other words, if you actually have 4 clear hours out of the 8 hours between flights - - then, yes, you could perhaps get a taxi or bus to Windsor. You wouldn't really have time to go around the castle but you could certainly have a walk around, maybe a bit of a walk on the Long Walk, maybe a walk along the river path, as long as you don't leave it too long to go back. It would be very, very tight to risk going into central London. Hypothetically it might be doable but you would be biting your fingernails down to the knuckle - or maybe you wouldn't mind another full fare ticket to Budapest...
Thank you for responding so fast. You have all answered my question. I will stay put until my Budapset flight.