Yes, an hourly coach service from South Wales to BRS was announced recently which will also serve north Bristol.
The Bath route will serve parts of south Bristol.
The Airport Flyer bus has been running for over a decade and operates 24 hours a day at 10-minute intervals for much of it, serving the central area hotels, country bus & coach station and Bristol's main central railway station (Temple Meads) as well as a different part of south Bristol to the new Bath service. The Flyer now carries over 600,000 passengers per year and has been steadily increasing its numbers.
There is also another bus service from central Bristol via the airport to Weston-super-Mare.
Given that BRS has no rail link and will almost certainly never have one the supplementary bus services are beginning to come together now to complement the core Flyer route.