Walk around London during Heathrow layover- itinerary help

Landing 11:00, out of immigration 12:00-13:00 (assuming plane on time), tube into central London ~1 hour = 13:00 - 14:00
19:30 departure for Vienna, plan to be at Heathrow 2 hours before (17:30), leave central London 16:30
You have 2½ to 3½ hours in central London, just do-able.

"Piccadilly line to Leicester station" - It is Leicester Square station. Leicester station is in Leicester, a city in the midlands.
"take the tube from Westminster station to Green Park" - nope, take district line to Hammersmith. It is a cross-platform interchange to the Piccadilly line.

Conclusion: rushed but do-able. You need to keep checking your watch.
What about your luggage? Is that checked through, or will you have to find somewhere at Heathrow for it?
Make sure you know terminal numbers. Different terminals have different tube stations.

Don’t. Sorry, you’ve got an awful layover but don’t leave Heathrow. Spend the money on a nice meal instead. Don’t add to the endless stress that is international travel.

Hi Sarah. Yes, great idea, go for it!! Don't worry too much about the detail or exact route of where you will walk. Feel free to wander if you see something interesting. Just have a tube map, and wherever you are at 4:30, get on the tube and head back to Heathrow. Enjoy!!

I think it is doable - we did something similar last September. You could shorten the walk and have time for a coffee in the park if you skipped Trafalgar Square. Get off at the Green Park exit - you can walk up the ramp directly into the park. Walk through the parks, heading towards Big Ben. You can see the London Eye across the river. Head over to the Abbey, then back through the park with a stop in front of the palace. Head back to the same Green Park station. The walk is about 2.5 miles and would take about an hour with no stops.

If you have plenty of time, you could start your walk from the Picadilly Station and walk through Trafalgar Square on your way over to Big Ben which is currently obscured by scaffolding.

I would do this. As long as you keep moving and nothing goes wrong you'll have just enough time for a whirlwind walk.
My first suggestion is to spring for a ticket on Heathrow Express...15 minutes from Heathrow to Paddington Station. Then the tube, Bakerloo Line to Piccadilly Line to Leicester Square (about 20 mins.). Walk down through Trafalgar Square, down Whitehall to Westminster, pop into Westminster Abbey for 20-30 minutes (keep in mind there may be a long line to get in...this is an extremely popular place to visit for obvious reasons! You may not have enough time). Walk back up Whitehall to Horseguards Parade, cut through the parade ground (or from Westminster Abbey you can just cut through the neighborhood directly north of it and hit the park from the south side), and then straight through St. James Park to Buckingham Palace. Then walk just across the north part of the park to Green Park Station...about an 8 minute walk. Take the Jubilee Line to the Bakerloo Line (be on it by 5:00) back to Paddington and catch Heathrow Express back to the airport. As long as you can be back on the Express by 5:30 you should be fine.
Bottom line...if you can be on the Heathrow Express, heading into town by 1:30 or 2 you can make it. If you can't then you shouldn't try. If you can only take the tube from Heathrow into town then you shouldn't try.
Also...download the app "Tube Map - London Underground" and use their Route Planner to get around. Sometimes there will be closures or delays on the tube and this will give you real-time traveling time as well as alternate routes if need be.
You only live once!! Go for it...

The Heathrow Express/Bakerloo/Piccadilly to Leicester Square won’t be faster than taking the Piccadilly line from the airport once you factor in waiting for the train, buying a tube ticket at Paddington, the long walk between the rail station and the Bakerloo platform, and the transfer at the very busy Piccadilly Circus.

Your plan to take the Piccadilly line the whole way will be the fastest (about 50 minutes) and most direct way.

One more thing ... check to make sure that there are no tube closures on the date that you travel
https://tfl.gov.uk/status-updates/planned-track-closures

There is a partial closure on the Piccadilly line the weekend of May 25th.

Thank you very much everyone for your replies! I really appreciate your info and suggestions, it's all super helpful :)

Sarah FWIW 8.5 hours in an airport is way too long to hang around. Looks like you have some reasonable options to do something else and maximize your vacation. Have a great time!

For someone who has never used the Tube it can be daunting and the maps aren't particularly helpful. I've watched countless tourists stare confusingly at the route maps (I'm talking the basic line map not the map of the entire system). Even I, who has used the Tube countless times, got on at the wrong direction last weekend and had to get off at the next stop, cross platforms and catch the next one back.

I'm not the most laid back of people so for me I would consider the options given to be far too stressful and as you are already anxious about the prospect I can't imagine the level of anxiety you'd be experiencing on the day! Personally I would opt for a quick visit to Windsor.

Is it really worth the risk of missing your flight to Vienna, paying a very costly walk up price for the next flight just to be able to have a quick glance at a few landmarks?

It seems eminently doable and you even have spare time if your flight in is delayed. Ignore the gloomy naysayers!

My gloomy suggestion is to have an indoors "Plan B" if the weather is awful and you won't want to be walking around outdoors so much.

Just chiming in to comment on your note that you have never traveled overseas before - as such, you don't yet know how you react to jet lag. Speaking for myself, I am a zombie after an overnight flight (I don't sleep well on planes), so I try very hard to minimize any stressful/high logic situations on my arrival day. That includes complicated transit (which you have planned), a hard deadline for a flight, and potentially many many things that could cause stress (delayed incoming flight, closed tube lines, airport security delays, busy disorienting city streets, etc.) I totally understand why you might want to do a whirlwind tour of London. But even with the best laid plans, for me it would be more stress than it's worth. London deserves more of your time. Consider a closer option (Windsor) or make the best of your long layover (here are some ideas: https://www.sleepinginairports.net/layovers/things-to-do-on-layover-london-heathrow-airport.htm).

I hope you are not flying Halifax/Toronto/London/Vienna?