Ytdyklly

This question will be badly articulated because I don't understand modern technology in the slightest, so please bear with me...
Assuming;
- there was no power, so there was no web/internet
- you had a mobile phone (one of the smart ones, not a basic one) powered by solar
- the phone had maps downloaded on to its memory card (so you were not reliant on the internet for map access),

could you still use satnav?
My rudimentary understanding is that satnav works through orbiting satellites constantly pinging down their location enabling you to exactly pinpoint where you are on the earths surface. Without the internet, could your phone still access this, or is the internet still needed?

No, the internet is not needed for a GPS-based device to calculate location.

However, the constellation of satellites requires regular maintenance and replacement. Without it, the constellation will begin to fail within a couple years, and GPS coverage will become increasingly spotty. There are multiple SATNAV systems in orbit - GPS was the first and is still the most popular. The others have the same requirement for regular maintenance and replacement.

Location means just that - a unique latitude/longitude spot on the Earth. Without a database of paths and other locations, that is of limited use to folks. Standalone GPS devices have the database pre-programmed. Phones download databases of varying sizes and uses from the internet.

There are easy alternatives: You can just follow the highway signs and use paper maps, the way everybody successfully navigated before 2000-or-so.

In addition to the maintenance of the satellite constellation there's another problem that's going to degrade your fixes unless you have a military device--the weather. Lightspeed is only constant in a vacuum and the upper fringes of Earth's atmosphere aren't quite a vacuum--enough to matter at the precision that GPS needs.

Normally, part of the information your receiver gets from the satellite is an update on this weather--but nobody's going to be making the weather reports anymore.

There is a second broadcast on another frequency that can be used to calculate and correct for the weather (the effects vary with the frequency, by seeing the difference between the two signals you can figure out how much atmosphere got in the way) but it is encrypted, if you don't have a military device you can't read it. Note that this is also why military devices are a bit more accurate than civilian ones--they get realtime calculations across the actual path the signal took, civilians get the periodically-updated weather reports that are an average over a large area.

Overall, the effect will be the fix degrades in a somewhat random/somewhat predictable way (a satellite that has drifted away from where it should be will make the same error every time it comes around--I think this will cause an error that repeats on a 12 day cycle but I'm not sure) and eventually starts getting periodic holes in the coverage as satellites die and sometimes you don't have 4 satellites above the horizon. Note that there are receivers that are built to look at all the satellites up there, not just the US ones. These will degrade the same but go a bit longer before getting no solution at all. (These units are commonly sold for polar use as the Russian system is optimized for high latitudes.)

First, we must assume that the GPS (or Galileo) satellite constellation is intact, and the satellites have not been shot down as part of the apocalyptic events in your world, and have not been subjected to EMP from nuclear blasts. If all is working up there, then they can continue to transmit their signals, which are nothing more than very accurate clocks (plus an almanac of where every satellite in the constellation is supposed to be).

Down on the ground, you have a device (smart-phone or standalone GPS) that is battery-powered, and a means of charging it. The complicated maths that allows GPS to work operates entirely on board this device. Comparing the time signals from multiple satellites and using the minute (ie pico-seconds) differences (which exist due to the speed of light) to derive a geolocation co-ordinate, is all done on-board your device.
Once the co-ordinate has been calculated, it can be compared against the index of a map stored on your device and voila, you know where you are. You can then use your choice of algorithm to calculate the best route to another location in that index, also done entirely on-board the device.

In summary, Yes, SatNav would still work. However, as has been said in comments and other answers, without replacement satellites, and continual updates to both the satellite orbits (and/or the almanacs they transmit) and to the maps on your device, the accuracy and usefulness of the navigation will degrade rather quickly, within a matter of years at best.