How to determine the Apple Watch model?

The WKInterfaceDevice.current().model property does not give a model number:

For Apple Watch, the value of this string is Apple Watch.

How can the exact Apple Watch model be determined from iOS?

There is no public API to get that exact information.

You can however use the following (I'll let you translate into Swift):

- (NSString*) modelIdentifier {
    size_t size = 0;
    sysctlbyname("hw.machine", NULL, &size, NULL, 0);
    char* machine = malloc(size);
    sysctlbyname("hw.machine", machine, &size, NULL, 0);
    NSString* model = [NSString stringWithCString: machine encoding: NSUTF8StringEncoding];
    free(machine);
    return model;
}

This returns a string in the format: "Watch1,1". You'll need to provide a lookup table to do ID -> Name translation.

"Watch1,1" -> Apple Watch 38mm
"Watch1,2" -> Apple Watch 42mm
"Watch2,3" -> Apple Watch Series 2 38mm
"Watch2,4" -> Apple Watch Series 2 42mm
"Watch2,6" -> Apple Watch Series 1 38mm
"Watch2,7" -> Apple Watch Series 1 42mm
"Watch3,1" -> Apple Watch Series 3 38mm Cellular
"Watch3,2" -> Apple Watch Series 3 42mm Cellular
"Watch3,3" -> Apple Watch Series 3 38mm
"Watch3,4" -> Apple Watch Series 3 42mm

By the way, this sysctlbyname API also works for iOS.

Cheers.