3 part question on travel in Britain

The Marriott website brags about walking to the station from the hotel which is in the Rugby stadium in ten minutes. The TfL planner says it is an 2 minute walk to Rugby Road / Whitton Road and then after waiting for a number 281 bus which runs every 8 minutes, then 11 minutes on the bus. The Marriott says the 281 stops directly outside. The walk according to the TfL is 13 minutes, the same as taking the bus, not counting waiting time. The Marriott says it is 20 minutes from the station to Waterloo. Not all trains are that fast. Looking only at trains making no changes, the 06:17 takes 54 minutes, the 06:34 takes 47, the 06:47 is 56 minutes. The xx:11 and xx:41 at many hours are as fast as 23 minutes but most of the fastest trains through the day take at least 26 minutes. I haven't seen any as fast as 20 minutes. If you only get to the platform 4 minutes before the train leaves, ticket in hand or using Oyster, take the one that takes 26 minutes and want to get to, say, the Tower of London I estimate it would take about 1 hour 10 minutes end to end.

Don't let anybody tell you that staying in Twickenham is staying in London - it is quite a remote suburb. How old are the teenagers? Under 16s are half price child fares on the trains. There is no Underground anywhere near Twickenham so you will be relying on buses and trains. Twickenham is way out in Zone 5 of the Transport for London price structure. There is a major interchange station south of London called Clapham Junction which you can reach from Twickenham (South West Trains) either via Barnes or via Strawberry Hill. Neither way is particularly fast as I remember, 8 stops one way, 10 the other. There are faster trains on that line and some get to Clapham Junction in only one stop. Be aware not to sit in the last 4 coaches of a 12 car train if you want to change at Clapham Junction, only the doors on the front 8 coaches can open there due to the short platform. Change at Clapham Junction for Portsmouth for the faster run, changing again at Woking; for fewer changes but a little slower change at Waterloo and come all the way back out again. Either way it will be between 2 and 2 and a half hours minimum. From Heathrow I'd taxi. Getting to Twickenham by public transport would be around the sun to get to the moon.

Marriott Twickenham boasts 20 minutes on train to Waterloo, Is this not correct?

Depending upon your departure time and number of stops, direct trains from Twickenham to Waterloo make the trip in 23-37 minutes. According to the Transport for London Journey Planner, you can take the Tube {Piccadilly Line} from Heathrow to Hounslow East, take a 7-minute walk to the Hounslow bus station, take bus 281 to RugbyRoad/Whitton Road, and take a 1-minute walk to your hotel. The total trip time is 42 minutes.

Jeff, do you absolutely have to stay in Twickenham or do you have some flexibility to stay in another location with better public transportation available?

No I just had a great deal there and it looked like a cool place to stay I have 2 boys heavily into a couple of sports . It looks like this is going to be too inconvenient , I am going to focus on the city . What particular area would be best suited for hitting the historic
sites?,

For an economical hotel in London close to the sights I'd recommend the Sanctuary House Hotel, http://sanctuaryhousehotel.co.uk, which is just 2 blocks from Westminster Abbey (you can look down from the hotel's front door and see it)and is also walkable to the Houses of Parliament/Big Ben, & a bit farther to Buckingham Palace. It has discounts on weekends like some other London hotels. I've stayed there & it's kept very clean. You could take a National Express bus from Victoria Coach Station to the port of Portsmouth for about £12 each and it takes just 1 hr 40 min. Since you're in Pittsburgh, US Air (via CLT) might be a good airline choice since they land at Gatwick, which is smaller and easier to get around than Heathrow.