Assuming your baggage is checked through, and you use the airside inter-terminal connection path, it would take you:

20 minutes to walk from your Term. 1 gate to the Connection Centre
15-20 mins., depending on queues, for security at the C.C. and check-in for your Term. 3 flight
15 mins. for Term. 3 connection bus latency and trip to T3.
5 minutes to navigate the T3 maze through to the departure lounge, or, as much as 15 minutes walk to your gate, if you're going directly there.

Total: from 55-70 minutes, depending. It can actually be faster to clear immigration, baggage, and customs, and push a trolley through the tunnel to Terminal 3, and check in again, but, depending on the airline, if you're in coach, the check-in queues as Terminal 3 can be very long.

In the other direction (Term. 3 to Term. 1), it's worth noting that the BA queue at the Connection Centre can be interminable, but there's no reason to wait there if connecting to a BA domestic flight since you can check-in at the BA domestic lounge in Terminal 1.