We arrive on a United Flight(Terminal 3) at 7:15am. Our British Midlands flight to Paris(Terminal 1) leaves at either 10:40 am, or 1:10pm. We would rather take the 10:40 flight.

Do we have enough time to transfer between terminals?

If we can get our bags checked through to Paris, it seems this will make it easier. But will we have to go through immigration/customs just to transfer terminals?

If your bags are checked through and if there are no serious delays, then the 10:40 flight s/b fine. You do not have to go through customs to change terminals.

What if our bags aren't checked through - will we still have enough time?

Even if you have to pick up your baggage you should have plenty of time barring major disasters. It's not that far between terminals. There is an inter-terminal bus but you can walk.

Thanks for the replies, hopefully we will be able to make the 10:40 flight.

Just to point out that transfer information is covered extremely well on the official Heathrow website which is called www.heathrowairport.com