Hi Benoît,
Had similar issues on the hardware serial using gpsd with ublox.
In my case gpsd switched to the wrong baudrate without knowing at which rate the unit is running.
try this command to start gpsd, it takes some time but both serial device as gps unit should be running at 115200 baud after that.
for SPEEDS in 460800 230400 153600 115200 57600 38400 19200 9600 4800 2400 1200 600 300; do ubxtool -w 0 -s $SPEEDS -S 115200 -f /dev/ttyUSB0; done; ubxtool -w 0 -s 115200 -f /dev/ttyUSB0; gpsctl -f /dev/ttyUSB0; gpsd /dev/ttyUSB0