Hot water upstairs is also fine. The problem is with the cold water upstairs. If I turn the tap on it runs at a low pressure for about 4 seconds and then just trickles to a stop. The toilet refills incredibly slowly as well.
The cold water tank is in the loft, and is filled to the top with water, so no problems there.
How long has the problem being occurring? A long time, getting slower over a time, or just now?
Have you checked if there is a blockage in the pipe run which supplies the tap? If you have an isolating/service valve before the tap, turn off the supply, disconnect the valve, connect a hose running out of a window/into a bath, turn on supply, check each section until you find out where it stops/where it runs normally.
What is the bottom on the CWSC in the loft like? Has it a layer of sludge on the bottom? If so flush it out. There should be a isolation valve on the cold distribution pipe, again disconnect, fit a hose pipe to run out to drain somewhere and swill the tank
You would think that if the ground floor is ok the upstairs would be alright as well, or has it always been like it?
How many cold taps downstairs? Is it just the kitchen sink or is there a loo / cloakroom as well? If just the sink the cold is probably direct from mains so that won't help. If a cold tap in loo what is the flow like there compared to a) the kitchen tap and b) the upstairs taps?.
"Hot water upstairs is also fine. The problem is with the cold water upstairs. If I turn the tap on it runs at a low pressure for about 4 seconds and then just trickles to a stop. The toilet refills incredibly slowly as well."
First it isn't a pressure issue - it's a matter of flow rate. Is it the wash basin and the bath and the loo or is the bath OK? What type of taps do you have - 1/4 turn ceramic discs type or conventional type? That may not matter if the loo is also giving a problem but it would be useful to know. Are the service valves for the upstairs cold taps and the loo fully open?
"The cold water tank is in the loft, and is filled to the top with water, so no problems there."
Hope you don't mean to the very top! That will keep you a standard pressure across all hot and cold taps fed from it (in case of hot via your HW cylinder but that doesn't matter in this case).
I'm beginning to suspect a blockage somewhere in the cold feed. Answers to questions will help.
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