Reading from EEPROM and writing to addressable LED with limited connections
My working details, per-request
Setup
So, the setup is the same as you what you have in its essentials:

The code is your code with the near-1 kHz tweak.  That said, I believe I also tested it at the default rate and that worked as well.  In any case, the images that show the LED changing were taken while that 1 kHz adjustment was made.

The UNO and Nano are functionally identical for this.  They use different serial transceivers which has nothing to do with anything.   The Nano will typically have a lower &quot;5V&quot; than the UNO when USB powered owing to the Nano using a diode where the UNO has a FET.  Which is unlikely to matter either.  They're otherwise the same device so far as the setup goes.

The 24LC02B uses the same pinout and I2C address and command-sequences as your AT24C16.  It is a smaller memory, not that it makes a difference; we're reading from address 0.  I did not find I necessary to add external pull-up resistors and it worked, so it seemed unnecessary to add them to demonstrate that it can work since adding them should only improve it's chances for working correctly.

The LEDs are the Adafruit ADA1643 12 Neopixel Ring. We're both using the first LED of a string, doesn't really matter how long the are.   We're only lighting one &quot;pixel&quot; so I just powered it from USB.  Have no idea what you're doing there.  But for one LED lit, I doubt a difference there would matter much.

You'll see a resistor connecting SDA at the EEPROM to IN on the addressable LED(s).  I put that there in case by some chance the EEPROM could drive SDA low while the addressable LED code was driving the LED data line high. while.  I doubt that can happen, but it was a small thing to do.  For what it's worth, I did run it briefly without it in series, and it also worked fine that way as well.  I'm only telling you this so you have all the information and not because I think it matters; I don't.


Running
It prints out 72.  This is the first value in my EEPROM. The 'H' of &quot;Hello world&quot; basically.  And then it goes on to cycle the first addressable LED of the set through all of the colors.
Configuration Pictures