As I understand it, the filmmakers co-operated with NASA - which is why the "Moonraker shuttles" look just like the real thing - and had originally planned for the film to coincide with the real launch, only for it to get delayed working out the kinks.
So, would your average British or American moviegoer in 1979 be familiar with the concept of the Space Shuttle, or was it more something you would need to be following "space news" to have heard about?
The Shuttle wasn't a "concept" in 1979, it was NASA's major crewed spaceflight program that had a budget of billions. Being the successor project to Apollo it had a lot of public attention, basically all the way from the last crewed Apollo capsule flights as part of Skylab and then the Apollo-Soyuz Test Project in 1974/75 up through the first launch of the vehicles there was continuous news coverage of the program.
In 1972 the first contract to build the test aircraft for the Shuttle program were awarded and by 1976 the first test vehicle designed for atmospheric flight was finished. This vehicle was carried aloft a 747 and then released at high altitude for glide and landing tests in 1977. It was effectively the first Shuttle Orbiter, though it was not a spacecraft nor capable of being used in a Shuttle Launch, though the loose plan of record was to retrofit the vehicle to be capable of spaceflight eventually. These tests and the preparation for them were heavily covered by the news media, so much so that in 1976 a massive letter writing campaign to the office of President Ford by Star Trek fans succeeded in getting the vehicle renamed "Enterprise" as well as much of the cast of the show (which had been off the air for years at that point) invited to the rollout ceremony. Unfortunately for the Star Trek fans it turned out that it would have taken extensive modification work to retrofit the Enterprise into a proper Shuttle Orbiter capable of spaceflight and instead a fuselage that had been used for structural testing (STA-099) was modified to become the Challenger Orbiter and the 2nd Space Shuttle to enter service after Columbia.
After the successful glide and landing tests the first Shuttle launch was originally scheduled to occur in 1979, but numerous delays due to issues with the heat shields, then the main engines, then a series of miscellaneous problems during attempted launches pushed that date back to 1980 and then to 1981.
By the time Moonraker had been released in June of 1979 the Enterprise atmospheric tests were already two years old and the first proper spaceflight capable Shuttle Orbiter, Columbia, had been built and ferried cross country on the back of a 747 to begin being processed for flight at KSC in Florida. All events that had received media coverage.
During development the Shuttle was heavily publicized and had a strong position in the public collective consciousness even before the first flight. The program was intended to not just be the next phase of crewed spaceflight at NASA, it was intended to be the end-all be-all future of spaceflight for America and the entire "free world". Not just a launch vehicle, but a highly reusable space plane that would fly regularly so often that it would dramatically lower launch costs and usher in a new space age. Before the realities of the deficiencies in the design and concept were painfully revealed through the progress of the program in the '80s through the early 2000s the hype for the Shuttle was very real and pervasive. It's difficult to properly impress how big of a deal the Shuttle was at the time. It was intended and planned to be the backbone of every aspect of spaceflight in the Western world, not just a crewed space vehicle. It was going to do everything, launch military satellites, launch space science probes, launch commercial satellites, retrieve or repair satellites (government and commercial), construct a space station, construct massive vehicles to facilitate interplanetary human space exploration, and so on. With a small fleet of Shuttles flying regularly it was hoped they could achieve dozens of flights per year.
The US government had bought in to that vision and funded Shuttle development based on those promises, while much of the American public (at least in the mainstream) did as well. Even during the development phase of the program the Shuttle was widely viewed as a major symbol of ongoing American primacy in space exploration (after the "win" of the Moon landing) and technological achievement in general. It featured in advertisements (here's one for an electronics company in 1979), in toys (there were Fisher Price and Matchbox Space Shuttle toys from 1979), and in popular media of course, such as the inclusion in the film Moonraker.
Another example of the Shuttle in popular media at the time was the TV show Buck Rogers in the 25th Century. If you watch the intro to the theatrical pilot (from 1979) you can see a space shuttle being flown by the titular character. While this is not the Space Shuttle, it is very much a vehicle strongly based off that design.
In short, yes, people were familiar with it, the public (via the TV and print news media) were well aware of the Shuttle Program, and it had a high profile in the public imagination even before the first launch.