If as you say the OBD also fails when the gauges fail, that might imply the fault is further back towards the ECU, or perhaps even at the splice joint. From the splice there appears to be separate signal wires running in parallel to the OBD and VDO.
Re soldering. That would be the best solution in principle but maybe tricky getting access, depending on the location. If it has to be soldered and things look fairly clean when the insulation is off may be try and solder the complete splice rather than the wires individually. Apply some flux first. You might need a substantial iron to get enough heat in too.
