You have a level translation problem. There are several potential solutions:
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Use an op amp to raise the level of the analog signal from ±2.5V to 0 to 5V. Disadvantage is you likely lose precision in the translation.
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level translate the digital signals in to the ADS1115. To do that the easiest way is likely optoisolators like this:
This is probably what I would do, as it avoids the analog problems. I’d use totem pole output opto isolators, although you may not need the speed (the versions shown will have a long rise time as it isn’t actively driven high.) You will likely also need adjustable voltage regulators for the +/- 2.5V supplies rather than the resistive dividers (that is likely a good idea anyway) because the optos will want about 20ma of drive current each for the diodes. The likely reason that the nano doesn’t see the ADS1115 is that a digital low is 0.7V from the nano, but that translates to +2.5V (because of the -2.5 ground reference on the ADS1115) which is likely above the threshold for a low level at the ADS1115. The 2.5V output from the ADS1115 is just barely above the high level to the nano which is likely about 2.4V (but may be higher) thus the level translators.
Peter