I was wondering about this... don't ask me why. GPS began in 1978. Were the specifications the same as today? Would a 1978 GPS device work today... and would a modern one have worked then?
Your limiting factor is selective availability. Prior to the year 2000, the GPS system was considered an instrument of national security, so the signals were intentionally degraded for other than U.S. Military use. Any device could listen to the signals, but only those with the proper codes could use them effectively.
There was a joke going around at the time of a grandfather and grandson going hiking in the Appalachian region. Grandfather had a topo map and a compass. Grandson had a GPS. During a break, they each took a moment to use their respective tools to fix their position. After a few minutes, the grandson announced, "According to this, we're on top of that mountain, right over there."
Modern GPS accuracy can be as good as 6 ft (2 meters) and is seldom worse than 40 ft. (12 meters). Before 2000 and without the security code to SA, it was usually no better than several hundred feet/ meters.
In 1979 GPS network was "in diapers" even for military use. There were only four satellites in orbit and that didn't give enough coverage or accuracy.
By then the satellite navigation system in use was TRANSIT, but you had to wait about an hour for a position or several satellite passes to increase accuracy.
In those years we used radio positioning systems such as Decca Navigator or Loran-C.
GPS began as a global positioning system with wide coverage (not global yet) and accurate positioning (for military) with enough satellites (10) in the mid-80s
But even then, for precise positioning you needed radio positioning systems as Syledis or Trisponder (local and specific use, and you needed to set up your own network)
So to answer your question:
Your actual cell phone would be useless as a GPS receiver in 1979, the existing GPS network wasn't enough for accurate position everywhere. Even with military-grade GPS receivers, was of not much use then.
It's likely that you would have to wait just a few more years. As of 1979, there were only four GPS satellites in orbit. Since a 3D position fix requires four satellites in view at once, the times and places where you could get a GPS lock in 1979 would be quite limited. By the end of 1980 there were six satellites; by the end of 1983, seven; by the end of 1984, nine; and by the end of 1985, ten. In 2016 there are 31 operational satellites. I'm not good enough with the orbital mechanics to say exactly at what point there would have been four satellites above the horizon more often than not, but it seems unlikely that it would have been possible with fewer than eight satellites in orbit, a milestone that was reached in 1984. Nonetheless, geodetic receivers (meant for non-moving applications where you can collect data for a long period of time and process it afterwards) were available as early 1982.
Compatibility shouldn't be a concern; the signals broadcast by the very first GPS satellites are still in use today and decoded by receivers. In the earliest years the system was a test platform, and its actual reliability unknown to me, but by 1982, with universities and government departments using it for geodetic work, you can expect that it would be somewhat reliable.
You should expect lower accuracy (due to Selective Availability, fewer satellites available to compute a fix, and lack of WAAS or internet augmentation), a slower time-to-first-fix (due to lack of internet augmentation), worse indoor performance (the first flight of satellites had a lower broadcast power), and worse "urban canyon" performance (due to lack of favorable geometry with fewer satellites). Also, map applications that rely on the internet for map data would obviously be out of commission. But given favorable conditions, I would expect your time-travelling smartphone to be usable for navigation from about 1982 onwards.
L2C only became operational in 2014 so all current receivers are likely to be C/A code compatible. As well as selective availability there is a possibility that modern receivers would get confused when operating prior to the 1999 GPS Time roll-over (overflow). Poorly designed receivers might also get confused by leap seconds.
The current standard civil signal, L2C, is only broadcast from satellites launched in and after 2005. The answer is no; GPS was extant, but the signal used was restricted.
EDIT: A bit of research tells me commercial products use a combination of the classic L1 signal and the civil L2C signal. Either alone would not be as accurate, however I don't know if a commercial product can use the L1 alone.
GPS reports your position (and the time) by calculating distances from at least 4 satellites. This crucially relies on knowing the positions of the satellites. As I understand it, part of the problem of a cell phone not being able to report its position when there is no network available is that it is attempting to download the latest "ephemeris", or satellite-positions information, and unable to do so.
I doubt it would be possible to get the ephemeris for the satellites of the period, but if you somehow could manage it you would then have to force the phone to use that arbitrary ephemeris. (Since even if your phone could connect to the towers using whatever technology was current, and you could establish an account with a provider of the time, that provider would not yet have protocols for furnishing the needed data.)
No it wouldn't work simply because the communication and data protocols have changed many times since GPS has been introduced. You wouldn't get any signal at all because your device and satellite wouldn't be speaking the same 'language'.