There's no nonstop path between Central Texas and the West Country, which is exactly why the connection you choose matters more than usual on this itinerary. Business class from Austin to Bristol requires a transatlantic hub transfer, and after years of routing through this corridor, I've settled on a firm conviction: American Airlines via its Heathrow or Philadelphia gateways, Delta through Amsterdam or Paris via its joint venture partners, and United via Newark or Chicago each offer viable paths, but the smoothest experience comes from pairing a comfortable long-haul segment with a short, low-stress regional hop into Bristol. American's widebody aircraft on the AUS-to-hub segments typically feature Flagship Business seats in a reverse herringbone configuration with direct aisle access, while Delta's transatlantic metal frequently rotates in its Delta One suites, complete with sliding doors on select A330-900 and A350 equipment. United counters with Polaris seating on its 767 and 777 fleets, a solid if less flashy product than its rivals' flagship suites.
The insider move here is booking the final leg into Bristol as its own lightly loaded short-haul segment rather than a single-ticket long connection wherever possible — this gives you flexibility to clear immigration at a major hub like Heathrow or Amsterdam, decompress in a proper lounge (American's Flagship Lounges at DFW or PHL are worth the layover time), and catch a later regional flight without the stress of a tight connection after a red-eye. Bristol Airport itself is compact and efficient, but it has no dedicated business arrivals lounge, so plan your last connection with that in mind rather than expecting a soft landing on UK soil.
Seasonality plays strongly in this route's favor if you're flexible: January through March and October into November consistently bring softer transatlantic demand and better award availability, particularly on the Delta and United joint-venture inventory that can be scarce in summer. Given how much these fares fluctuate hub to hub, I'd strongly recommend setting up automated fare tracking for this specific city pair so you can pounce the moment premium cabin pricing dips into a reasonable range.