What version where you updating from? Did you get an "OK" response in the update window e.g
When you power cycle the unit do you see any WiFi related messaged open the openevse LCD? The WiFi module should print it's version number and local IP address to the LCD at statup.
Have you checked to see if the unit is an AP mode? Can you see any SSID with openevse?
If unit seems dead then you will need to upload via the programmer You can upload the .bin directly via serial using esptool. See instruction on the readme
I was updating from 2.5.2. I did get OK, and then I hit restart. While plugged in, the red led is constantly solid.
There are no messages on the openevse LCD. I have checked for an AP with SSID broadcasting, but nothing.
I suppose I can't use the programmer I have for the openevse board that I mentioned above.
Mmm that's strange. Before re-flashing you could use the programmer to see if you can read any serial from the unit @115200 baud.
If you are just communicating with the openevse board using a programmer then this programmer will be fine to use. You want a serial to UART or FTDI programmer. Not an ISP programmer. This programmer is fine to use: https://openevse.dozuki.com/Guide/Serial+Communications+with+OpenEVSE/13