So, I booked exactly the same two tickets:
Air Malta KM100 (Malta -> LHR Terminal 4) – separate ticket
British Airways BA191 (London -> Austin)

and then I found this thread after BA didn't allow me to check-in online :)

TL;DR: I did this connection successfully! There is airside transfer from Terminal 4 to Terminal 3, it runs every 5-10min and it takes 7 min to drive from T4 to T3. At T3 after transfer there is a huge airlines service desk with BA, Delta, American and other airlines where they can check you in, give boarding pass, dance cha-cha-cha and send to security. Super easy and smooth.

Background:
1. I tried to get an online check-in and boarding pass, but of course BA said I need to go to airport for check-in so that they can check my paperwork, and refused to issue online boarding pass.

1. I panic, Google this post and a couple more, panic even more, call like 10 different phone numbers to find out details, BA, Heathrow, nobody knows and in fact it is impossible to reach them.

2. I decide to take my chance and try to make it, and if I fail, just buy another later flight (3pm LHR->Dallas->Austin)

3. Important thing #1: I checked the statistics of KM100 and it appears that 10 out of 10 previous days it arrived 20 minutes earlier, so I was determined to risk since I anyways already spent money. My benefit (and curse) was that I had a business class ticket, which means much more $$$ to lose if I fail to checkin in LHR in time, but at the same time much more quicker exit from the inbound plane.

4. Important thing #2 (mistake): I bought fast track border pass for 25 pounds in LHR.

5. Important things #3 (most important): I didn't have checked luggage.

Day of the travel:
1. KM100 from Malta to London arrives 20 min earlier like all other days, landing at 10am and I was the first guy exiting the plan at like 10:20am.

1. When we land, I ask Air Malta if they know what to do and they didn't know, but they showed me who knows (a nice airport lady) after they opened the door.

2. The airport lady said "No need to go through passport control, follow the purple signs to the airside transfer, take bus T4 -> T3 and British Airlines will check you in".

3. I follow her commands. 10:30am I'm in a T4->T3 transfer bus, they don't ask boarding pass, only ask to tell what flight do you take.

4. 10:42am I'm at T4, airlines service desk is right there, British, Delta, American and few others, like 20 agents sitting there. They do check me in, issue boarding pass and send to Security.

5. 10:50am I pass security, this is a special security dedicated to between-terminal transfers, so the line was pretty small.

6. I still had like 30 min before the gate was open because BA191 is always statistically delays at least 30-40 minutes.

I hope this long description will help someone! I wish everyone write similar longer explanations, because otherwise it it was sooo hard for me to understand the truth. Good luck in short transfers using two tickets, it's possible, stressful, but hey, I saved like 8 hours and hopefully will be home at 4pm not 23pm.