Any ideas why one upstairs toilet cistern will not refill properly each time it is flushed?

It doesn't look like the water is flowing through the cistern, at least after flushing i cannot see water flowing, unlike my other toilets.
Also, if i turn the water off under the cistern, then turn it back on straight away, the water starts flowing immediately. And i noticed that if i run the water in the sink or shower in the same room, that it sometimes makes the water flow properly in the cistern.

Is the float arm (ball valve) falling when the toilet is flushed? if it is, is it going all the way down or is the valve sticking closed?

If neither of these is the problem, and from your description I don't think that either of them are, there is probably a piece of solid material like rust, solder or broken off calcium deposit acting as a valve at some point in your water inlet. It must be somewhere narrow like a bad bend or a stopcock or the actual inlet valve. When you cut off the water it falls down the pipe out of harms way, when the water has been flowing for a while it gets swept back.

As you can easily cut off the water to your cistern, turn it off, dis-assemble the inlet valve if you feel confident, it isn't difficult (or even better, get a replacement of the same type). When the valve is off, turn the water back on to flush the pipe, you will have to cover the valve with a jug to stop water spraying everywhere. Re-assemble the valve, having replaced the washer or replace the whole valve. If it's not fixed, the blockage may be in the actual stopcock you use to isolate the cistern.

If there are a lot of knocking noises, it may be an airlock. Is the cistern fed from the mains water supply or from a tank in the loft.

first thing I'd do is replace it. Then I would run my checks.