Ampere-hours [Ah] is a charge. It says nothing about power, so you cannot determine your cable sizes from that. Batteries have a "12 Voltage, 120 Ah capacity" value. Don't mix that up with amperes [A]. They are not the same.
Your 1500W inverter will draw up to 125A from 12V, or 62.5A from 24V. It is generally adviseable to have lower currents because of smaller losses and easier construction, as long as you don't leave the low voltage range. (I would even advise to get your hands on a 48V inverter and battery set to drop the losses even lower.)
A rule of thumb is at minimum 0.1mm² of cross section per ampere of current. The lower the system voltage and the longer the cables, the more cross section per ampere you want to reduce the voltage drop. I recommend at least 0.2mm²/A for the 24V system, and 0.3mm²/A for the 12V system. (Same: don't mix up cross section [mm²] with diameter [mm]. They are not the same.)
So, I recommend:
- 4mm² for 31.25A@48V
- 12mm² for 62.5A@24V
- 35mm² for 125A@12V
Short cables may be much thinner.