Interestingly, the test light comes on slowly and is quite dim at the fuel pump.. When at the battery it is bright and immediate.
Knowing the exact voltage is a useful test in diagnosing. Also testing with a test light is required for the above reason. You can easily have full voltage but not enough current flow to power the motor due to a bad connection in the circuit somewhere. The volt meter will still register full voltage, but the test light will burn dimly indicating a problem.
As reference test, connect the test light directly to the battery positive and negative posts so that you will know what your test light looks like when it burns at full brightness.
Disconnect the wiring harness from the fuel pump.
Connect one end of your test light to the fuel pump connector 12V+ terminal, and the other end to the fuel pump connector 12V- terminal. This tests both the 12V+ and 12V- sides of the entire circuit all the way from the start to the fuel pump connector at the same time. The test light should burn as brightly as it did in the reference test. If it does, the electrical side of the fuel pump wiring is probably not at fault. Check the condition of the connection between the wiring and the fuel pump, and the fuel pump itself.
If it burns dimly, there is a fault in either the 12V+ or the 12V- side of the circuit.
With the wiring harness still disconnected from the fuel pump;
To test the 12V+ side of the circuit, connect one end of the test light to the battery negative post, and the other end to the 12V+ terminal of the fuel pump connector. The test light should burn as brightly as it did in the reference test.
To test the 12V- side of the circuit, connect one end of the test light to the battery positive post, and the other end to negative (ground) terminal of the fuel pump connector. The test light should burn as brightly as it did in the reference test.
If the test light burns dimly when testing either the 12V+ or 12V- negative side of the circuit, that is the side of the circuit that you need to investigate.
Because you wrote that the light comes on slowly, that indicates that there is a problem with the relay circuit. You tried another relay with the same result. While it is possible, it is unlikely that you have two defective relays. Assuming it is not the relay, your next step would be to test the circuits that trigger and feed the relay at the relay connector using the same method as above for each wire. You can also jump the wires at the relay. This eliminates the relay, so if the fuel pump works you will know that the wiring from the relay to the fuel pump are good.
You will need a wiring diagram to know which wire is the supply, which is the trigger wire, etc..