Just take the earliest flight and get to your destination and have a shower once there would be my recommendation if you were booked on a through ticket
Being on separate tickets, then you have no protection should you miss your onward flight and could be up for paying for another ticket from London to Paris. In this instance I would suggest heading straight to Terminal 5 and using the facilities in the 1st class lounge there. That way you are through security etc and no worries on how much time it will take to clear security from landside in Terminal 5
Dave