I had a very similar situation a few months ago. I was arriving at 5:55am on UA into T3 at LHR and departing at 6:55am on a BA flight to Berlin from T1. I thought I could make it (I had no checked luggage and I can run pretty fast!). The UA flight was 15 minutes late, and I got to the BA check in desk in T1 at 6:45am. They wouldn’t let me on, but they were nice enough to re-book me on the next one at 8:30am for free even though I was on a cheapo fare.

Long story short, an hour is not enough to transfer between terminals at LHR!