At it's core - whichever city has the strongest demand and/or inventive package for new service. See below for the long-winded answer.
Although AUS may not be the same size as PHX, it's growing at nearly twice the rate, fueled largely by the tech and business booms in the city. Given the nonexistant capacity in the AUS-International market, it's an attractive opportunity for a carrier like BA to enter and try to capture the market as it grows, before anybody else does. Couple these facts with an exceptionally attractive incentive package from the AUS (read: revenue guarantee), and it becomes pretty close to a risk-free opportunity. My guess would be that load factors and fares are strong enough in LHR-AUS that BA wants the extra cap for margin expansion - the 777 upgauge brings a nice decline in unit costs (say, 25% hypothetically - would need to see the actual numbers), while they can stimulate incremental demand with lower fares, in the hopes that their unit revenue decline is less than the unit cost decline (say unit revenue goes down 15%). Unit costs down 25% + unit revenue down 15% = margin expansion of +10 pts over today's service. This is all assuming, of course, that they can pick up the incremental traffic. If the upgauge hurts your RASM more than helps your CASM, it's all for not.
From the airline perspective, the major drivers when evaluating a market would be:
1. Passenger/fare data: how strong is the local O&D market, and what are the fares (or daily revenue) in the market? The latter is largely determined by capacity. But having a strong mix of business and leisure traffic, coupled with the numerous events over the course of the year in AUS, make it a great market.
2. How much capacity does the market have? PHX-LHR already has 1x daily BA service - on a 744, no less - so any incremental service is likely to saturate the market. The only way for a new carrier to make it work is either: a) stimulate new demand in the market with low fares (probably unsustainable for larger carriers, this is more of an LCC play) or b) shift share from the incumbent carrier with a combination of lower fares, superior product, and/or better schedules (this is what DL will try to do in PHL-LHR with their growing FF base on each side). If PHX were a huge market, Airways/American would certainly be flying it. Probably not a good two-carrier market.
3. Equally important, when deciding on several options at once, is who puts forth the best incentive package to mitigate your risk. As an airline, you're taking all of the risk on new markets - and, in the case of LHR-AUS, you might be building the market from something close to scratch. Without an incentive package, the market will hemorrhage money as the carrier is trying to build awareness and get folks in seats. If the airport team is willing to help offset some of these losses in the early years, you stand a much better chance at sustained service in perpetuity. My guess would be that of all the cities in North America that BA would consider for new service (RDU, CLE, PIT, CMH, etc.), AUS presented the best incentive package.