If both keys were somewhat working before the key batteries were swapped and then both keys stopped working after new key batteries were installed, then I’d suspect the batteries were not installed correctly. So open up the keys again and check the battery polarity is correct (there should be a printed or embossed + on the key half holding the battery.
FYI for others, below is how you change the battery
If that battery was installed incorrectly (easy to do when in a hurry) obviously they will not work. If the battery was installed correctly and it still doesn’t work, it gets trickier.
I’m not sure whether the car itself gives any indication that a key signal is actually being received as this would indicate both the key and the vehicle are ‘talking’ and that the key code is not being accepted. If so it could be that the car needs to be coded to accept the key. If no signal is being transmitted by the key or no signal is being received by the car, then it’s again either key of receiver issue. This is where it’s helpful to have diagnostics equipment that connects to the car but such stuff is costly
Anyway, in the VF ute, the key receiver is behind the passenger and under the headlining near the rear window. Could be in the same place within your VE Ute. But changing modules unless you’ve determined what is faulty can be an expensive diagnostics method.
As an aside, in the VF, there is a key add procedure that could be performed simply and without special tools to code a new key using both old keys. It works by inserting old key and turning ignition on, wait 10 seconds then remove key and insert second old key and turn ignition on for 10 seconds then remove key and finally insert new key for 10 seconds before removing it. At this point new key should be coded and can start the car. Maybe there is a manual key code process for VE you could try?
Probably more effective to go to a dealer and get it diagnosed. On that front, I have read somewhere on these forums that there is a procedure for starting the car with a key that sets off the alarm. I think it’s something along the lines of putting key in barrel and leaving it in for hours before turning on (but not really sure).