Does iOS know the phone's location solely from cell tower, i.e. without GPS data?

Your hypothesis is conceptually correct. GPS is a one-way system, with your phone listening to satellites broadcasting hyper-accurate timestamps from orbiting atomic clocks. A regular GPS unit can take 10+ minutes to download the latest satellite positioning information (almanac and ephemeris) needed for triangulation. This can lead to very long delay before having an accurate position, which industry folks call time to first fix (TTFF).

By combining the data connection of a smartphone and information about surrounding cell towers, the TTFF can be greatly reduced. This strategy is known as Assisted GPS or A-GPS. If you search around, you can find lots more information about A-GPS because its development was spurred by the Federal Communications Commission. They wanted to improve the routing of 9-1-1 calls (see Enhanced 9-1-1), so that if you call 9-1-1 from a cell phone you are connected with a nearby dispatcher, instead of one in your phone's area code (which could be across the country).

The A-GPS does not (officially) utilize Apple's servers. Instead, A-GPS is implemented by the carrier who owns the tower (e.g. AT&T, Verizon, T-Mobile), because the FCC wants this functionality to work with any capable device, not just Apple devices. This location information could later be disseminated to Apple or other websites and apps to which you grant access to your location information, but that's separate from A-GPS.

There are other systems that supplement GPS and A-GPS. The article that @Lernkurve linked to goes into this in a great bit of detail. These positioning services may be conducted on Apple's servers. A few years ago, there was actually some controversy about how iOS "accidentally" retained this location information without notifying users. There's also concern about the reverse being used, e.g. WiFi hotspots tracking your location via the phone's MAC address. Fortunately, iOS8 implemented some MAC address randomization to avoid this.

tl;dr

Does iOS know the phone's location solely from cell tower, i.e. without GPS data?

Yes, iOS does, using Assisted GPS (Wikipedia, dmahr).

How it works

The article Deeper insights into Apple’s “GPS” explains how it works:

All iPhones scan their environment for WiFi access points and cell towers and their signal strengths. If there is GPS reception at the same time, the iPhone can make a list of all found WiFi access points and cell towers including their approximate GPS position.

That list is sent to Apple. So Apple has a large database of WiFi access points and cell towers including their GPS position.

Now suppose that some other iPhone is in an area currently without GPS reception (bad weather, interference or something). So it scans its environment and finds some cell tower. It sends the cell tower's ID to the Apple via Internet.

Apple looks up this cell tower's particular position and sends it to the iPhone, and, voilà, the iPhone knows its position without GPS reception.

Apple also sends down WiFi access points and cell tower location data for the iPhone's current neighborhood, so the positioning will also work if there should be no Internet connection for a while because part of the database is now locally stored on the phone.