How have you reached your timings? They appear to be very tight and take no account of potential delays. How are you intending to travel to and from Windsor Castle? What terminal are you flying from? How long have you factored for immigration? Have you accounted for traffic? What about the security lines when returning to Heathrow? Have you considered how far your departure gate is from security? Will you need to walk a distance or even take the shuttle?
Your landing may be scheduled to land at 13:20 however the flight might be delayed or there may be significant taxiing time to the stand or even having to get a bus to the terminal which could add 30+ minutes. Then there's the walk to immigration and then immigration itself which probably would take an hour or more. Realistically you're unlikely to be leaving Heathrow until after 15:00 and factoring in getting a taxi, dealing with Heathrow traffic and potentially the horrendous M25 (if the driver opts for that route) then there is every likelihood that you'll miss the last entrance time.
For me your timings are far too concise and optimistic. Personally I like to add an hours buffer to the recommended airport times to cover potential delays and unforeseen events, it makes for far less stressful travel. If it was me, I'd be staying in the airport. If you don't have lounge access I would recommend buying it and just while away those few intervening hours taking advantage of the food and alcohol whilst reading, browsing the internet etc.