Read here somewhere that an EEPROM Write takes 3.3ms. Guessing that a Read is much faster?
Also, not clear how much EEPROM in an Uno or Nano. Some say "1024 bytes on the ATmega328" and elsewhere it's "512 bytes".
THis is helpful...
https://www.arduino.cc/en/Reference/EEPROM
Typically, on the AVRs, you have half as much EEPROM as RAM, so both the Uno and Nao, with 328 processors, have 2k RAM, 1k EEPROM.
If in doubt, check the processor's datasheet.
When the EEPROM is read, the CPU is halted for four clock cycles before the next instruction is
executed. When the EEPROM is written, the CPU is halted for two clock cycles before the next
instruction is executed