Check cold water feed
All homes in the UK receive a cold water "feed" from the water board. You would normally find that under the sink or in a downstairs toilet. It will be obvious as there is a tap (stopcock) valve that lets you turn on/off the fresh water flow into the home.
The more you open the stopcock (turning counter clockwise) the more pressure you receive, the less open (turning clockwise), the lower the pressure.
If you, or someone in you home, has turned this off or down recently; then the slowing of flow may be due to that. Simply open the valve/stopcock fully and then turn it back 1/2 a turn.
There could be an airlock in the pipe work, water usually starts, slows and splutters. If this is what you experience, turn the tap on slightly until any air locks escape, takes just a few seconds.
If you have a cold water tank in your loft, check whether that is full. If not that will be a reason why your tap is slowing and stopping.