Currency movements - Question

If the $ goes up against the £ - the UK is less likely to buy from america, and will buy from the japanese guy - causing an increase in the value JPY.

currency exchange rates are determined by demand. Either demand falls absolutely, or it shifts to another currency.

That's not necessarily true. Recently, when markets are avoiding risk, in general the Dollar would rally against most currencies, but fall against the Yen. A lot of it has to do with risk-tolerance, the carry trade (interest rates), volatility, economic conditions, etc. There is no hard and fast rule.

The only certain relationship is between 3 currency pairs like so:

EUR/USD, USD/JPY, EUR/JPY

Because you can synthetically create all the pairs, ie form EUR/JPY by Long EUR/USD, Long USD/JPY.

In theory exchange rates are going to be determined by a combination of purchasing power parity, covered interest rate parity and speculative demand. But to actually see the effect you are asking about you should look at the EER (effective exchange rate) which looks at a weighted average exchange rate for a country using a basket of other currencies. That shows you if it is moving up or down relative to all other currencies.