It might or might not help, but this is what I did.
Load the example sketch ‘Hardware Serial Parsing’.
Turn echo on:
// Set GPSECHO to 'false' to turn off echoing the GPS data to the Serial console
// Set to 'true' if you want to debug and listen to the raw GPS sentences
#define GPSECHO true
Adjust the baud rate for the GPS device. This is probably required because I have changed the default of my particular device, and should not be required if the default from the factory really is 9600.
// 9600 NMEA is the default baud rate for Adafruit MTK GPS's- some use 4800
GPS.begin(115200);
Open the serial monitor window and upload the sketch. This was the result:
Adafruit GPS library basic parsing test!
PMTK314,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,028
PMTK314,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,028
$PMTK220,10005Q-⸮0531
$PMTK220,10005Q-⸮0531
The echoed line tells me that the GPS device is working and talking to the MCU at the Serial1 port with the correct baud rate. If you get nothing then there is likely a problem in the soldering. If you get garbage (eg ⸮Q-⸮14,0,1b, ,0,0,0,0,0, b⸮b⸮b⸮R⸮⸮h0P)⸮⸮b⸮⸮⸮⸮R⸮25 ⸮Q-⸮14,0,1b tA⸮⸮⸮R⸮⸮j tA⸮⸮⸮R⸮⸮j) then the baud rate is wrong - try something other than the default. Once you get that message then you know you have communication and the problem is either that the GPS is not being initialized correctly or it is not receiving a signal.
Note that this test was with equipment somewhat different than yours, but it should still be valid. I will try to find some parts more similar to your setup and repeat this tomorrow.