About a month ago my water shut itself off due to low pressure. It came back on by turning the switch on the electric box connected to the tank. ( No t the fuse). After some debate we replaced that unit. During that install we noticed the tank itself was not filling itself. It was primed and filled but it did not last. Once the initial water was gone the pump continued to come on every time I turned water on and the tank remained empty. So we replaced the tank.
Unfortunately that did not help either. The tank was filled when installed but once that water was drained it does not fill up again and the pump comes on everyt ime I use the water.

I've also noticed that the pressure shut off can happen throughout the night, which implies to me that there is a loss of pressure going on somewhere even when the water is not in use.

The fix is easy enough, I just flick that switch but it's going to take a toll on my pump so would really like some advice before I burn it out.
I've read that when a tank is replaced it can take a few weeks to resolve itself and fill with water. Should I wait it out or do I still have a unresolved problem on my hands ? My next step would be to take a look at the foot valve in the well but after trying unsuccessfully to DIY, I'm debating if it's time to call a plumber.

You are correct about the tank. There is an amount of air in the tank which, over time, will be absorbed by the water and be gone.

Watch your pressure gauge while no water is being used. It should stay rock steady. Check it for an hour or two. If pressure drops, even slowly, then you are dealing with a leak somewhere. That is step one for you.

The fact that the switch is being tripped at night is strange. This happens whenever pressure drops to a set point which is typically very low. The switch "thinks" that the pump is running dry and so disables itself until you reset it. Is this also happening during the day?

You speak about a "tank". Is it a bladder or a pressure tank?(see images)
In any event nether one will fill completely. You don't mention having to prime the pump. Let's check the foot valve. Turn the power off to the pump and let it set for a few hours. Did the pressure gage drop? Now turn the pump back on. Did it start pumping or did you have to prime it?
If the pump pumped up to pressure then chances the foot valve's OK.
However, If the gage fell then you have a hidden leak either in the line going to the house or in the house itself. Are you on a slab?
jlisenbe gave you great advice. Just thought I'd put in my two pence worth,
Good luck, Tom