Kate,
Believe it or not, even with a "6:30am" arrival, you may not be free and clear on the "land" side of the airport until 9:30-10:00; especially if you checked bags versus carrying them on. If there are any flight departure delays (there will be) or there is any "weather" in London (there will be), you will probably be at least 30 minutes (likely more) late arriving. It's just the nature of things these days... I have not had an on-time arrival into Heathrow yet...
There are a surprising number of international arrivals hitting LHR at that time of the morning. Queuing for Border Control can be quite lengthy. If you check your bag, the wait can easily be 30 minutes or more.
If you decide to take the tube, there's the time needed to get a ticket and then a wait for the train. If going into the city (you didn't specify where in London you were staying), the fasted option to Paddington is 30 minutes while the cheapest takes closer to an hour. Then you still need to transfer to whichever tube line gets you to your hotel...
With that "0630" arrival, don't expect to be at your hotel much before noon. Chances are, as stated previously, you can drop your bags at their "left luggage." Then, you can go find some nice activity to help in fending off your jet lag.
As for Windsor Castle... It's a marvelous place to visit but it can be physically and mentally demanding. I recommend being well-rested before your visit.