Baskaran,
You don't indicate from where you are arriving from so I'll assume you are arriving on an international flight.
It will take at least an hour, possibly two, to navigate border control and custom (especially if you have checked bags). It takes a minimum of 30 minutes (Heathrow Express) to over an hour to get to Paddington on the tube (fastest way). From Paddington, you still need to travel to wherever you plan to sight-see.
I would stay in the city for the convenience of sightseeing. Most hotels will store your luggage for a few hours after check-out (on the 13th).
You will need to allow for travel, check-in and security time on the 13th so you really only have about 8 hours of sightseeing time.
I'm not a big fan of the Hop-on/Hop-off buses in London. Traffic is so bad and convoluted that it just doesn't seem worth it. I recommend getting a tourist map of the city and just walking or taking regular buses or the tube between sites. Just my opinion...
If it were me, I would stay at a hotel close to Paddington, check in on the 12th to drop my bags, and walk across Hyde Park to Buckingham Palace and fan out from there. On the 13th, Check out, store you bags at the hotel reception, hit the sites you missed the day before and then pick your bags back up on the way to Paddington for your flight to Manchester.