I cut the 3v3 trace on the backside of a Nano 33 BLE. I reconnected the USB cable to be sure it was cut and the power on LED didn’t turn on as expected. But now I can seem to get the Nano 33 BLE to turn on with an external battery pack, which I have connected via an LD1117 that outputs 3.3 volts - I verified that with a multimeter. I also connected the multimeter in series to check the current which is reading 16mA. The batteries I’ve tried are 9v, a 3.7 LiPo, and a pack of 4xAA.
I temporarily resoldered the jumper to make sure I didn’t inadvertently toast the board and it worked fine via the USB cable again. So I removed the solder but the issue remains: can’t seem to power on via the battery.
Also, I checked the voltage at the +3.3 pin (the pin right near the Aref pin) and it reads 3.3v.
The voltage checks out at 3.3, the current draw seems ok at 16mA and I temporarily reconnected and then disconnected the 3.3 trace but it won’t power on.
...I’m assuming that the onboard LED will light up when under battery power without the USB cable connected.
The LEDs are under software control. If you have a look at the schematic both LEDs are connected to I/O pins.
Is your sketch using the LEDs?
No, the sketch isn't using LEDs. Maybe my mention of an LED was misleading. What I was referring to is that the green LED on the board that lights up when the board is connected via a USB cable to my computer does NOT light up now that I've cut the 3.3 trace and connect a battery to Vin and GND. In other words, with the 3.3 cut, it doesn't seem like the battery is supplying power to the board yet that battery is providing 3.3v.
adr2018:
... connect a battery to Vin and GND.
There is your issue. The Vin pin is connected to the input of the MPM3610 Step-Down converter. You need to connect the battery to the +3V3 pin which is connected on the other side of the bridge and provides power to the MCU. The bridge is cut so that the power can no longer flow back to the output of the MPM3610. Have a look at the schematic of the board (lower left corner). You can download it from the store under Documentations -> Schematic in PDF.
You can also see what I wrote in the first post, the green LED is connected to an I/O pin. It might be that the bootloader switches the LED on but then it might not if USB is not connected. I do not know what the behavior is, just so you are not discouraged when the LED does not switch on. 
Use something from your sketch to determine whether the board is running. This has been designed to allow you to get minimum power with the Nano 33 BLE, by disabling it. On other boards this is not the case and some LED is always on when power is on.