You'll find out when you check in whether they through check your bags!
Anything is possible at Heathrow, such as flights running on time, security being easy etc. However, you will be pushed to make your connection. I suggest you wear some good, comfortable running shoes...
Edit - Found BA's transfer information (it's hiding under "Connecting at Heathrow") - they (well, BAA) suggest 2 hours connection time between 5 and 4.
Perhaps pack a folding bike in your carry on too...