Because things are often done differently at different places we have to be very clear about how and why things are done.
First tell us why this is being done this way, if you know.
Our house has a large fresh water tank filled by mains under the driveway
I assume that you mean that the tank is under the drive way not the mains.
A pump at ground level pulls cold water from the tank and pressurises the cold water supply to the house.
Are you certain about that statement. Where is the pump? Is it between the mains and the tank or is it between the tank and the house? That question is ask in the terms of piping, not physical location.
What kind of controls are on the pump? There should be a pressure switch to tell the pump when to run and a pressure gauge to tell you what is going on.
If I run a tap upstairs for a period (20 - 30 minutes ) the pressure will come back.
Does the same happen if you run a tap downstairs?
I am assuming that air is getting into the system?
In a pressurized system you don't have air in the system. If you do get air in the system, because air is lighter than water, it comes out as soon as you open a tap.
So tell us how things are done as best you know and why. Keep what you say short and simple. I am a simpleton, you would be surprised how many ways I can interrupt what you say.
I have preconceived concepts about how things are done and why. To express them now would only contaminate the conversation.
So confirm that the tank is under the driveway. How large is the tank? Tell us the timing and what the circumstance are when the low pressure occurs. Does opening a downstairs tap restore pressure like the upstairs tap. When you experience a pressure loss what is the pump doing, is it running? Do you have a pressure gauge and what is its reading?
The plumber the landlord sent initially tried to tell me the pathetic dribble of water we get was normal
Perhaps because you are on the other side of the world everything is upside down but statements like that are universal. When we get to Mars there will probably be someone or something there saying "Oh, that's normal".