For example, if you shower for 10 mins it will be fine, then the hot
water will just totally stop. Then 1 minute later it will start again,
but stop after 5 seconds, then stop-start constantly. It's as if
there's not enough water getting into the tank or pump.

Assuming you mean the water flow stops, not that it goes cold, I would
strongly presume that:

1. You have a gravity fed hot water system using a cylinder and a cold water
tank in the loft.
2. The bathroom is pumped, but the kitchen bypasses the pump, so has
inadequete pressure/flow
3. The pump is so efficient at blasting you in the shower, the cold water
tank empties and the pump stops.

Assuming this is the case, you'll need to ensure the kitchen is put onto the
pumped system (could also be another problem, like pipe blockage). You'll
also need to sort out the cold water tank situation. Possible solutions
(either individually or in combination) assuming to wish to basically keep
the same type of water system:

(a) Bigger/additional cold water tank.
(b) Ensure all valves to refill valve are fully open, including stopcock.
(c) Replace refill valve with wider bore type.
(d) Remove flow reducers from refill valve.
(e) Install multiple refill valves to increase flow.
(f) Replace rising main to refill valve with wider bore
(g) Fit a flow reducer on the shower, so the refill valve can keep up with
the flow.
(h) Move cold water in bathroom from gravity to mains, to allow all stored
water to be used for hot only. (May require different type of pump if
currently twin impellor).

Christian.