Everything else is just supply and demand, ie what's the market-clearing price for 1 BTC and for 1 BCH and how does it compare to before.
However, this is not always the case - due to people thinking they can get 'free coins' from the other chain (price goes up), people being afraid of a fork (price goes down), etc
I guess the closest comparison to bitcoin is; oil price is at $60 / barrel today and speculators believe there is a 30% chance that they mis-estimated the number of barrels by 100% - but you won't know for sure for another 3 months. What's should the price of oil be tomorrow?
When a cryptocurrency forks, you keep the same amount of coins on both sides but the 2 are wholly incompatible with one another.
A closer comparison is that you it's like 2 simultaneous futures, but with different consensus rules.
There was an interesting situation after the fall of the Soviet Union. Because most of the states didn't have economies strong enough to support their own currency, they remained a unified currency union as part of the CIS
The only problem was that each state then started printing the money like mad and then sending it to other states to be sold / traded. Some of the states then introduced what was a quasi two-currency system - to buy local currency or to exchange it, you required both the old Soviet Ruble and a new "permission" note from the state to exchange.
In theory this placed a cap on the inflation but the result was almost like a currency fork. A lot of these states ended up establishing their own currencies but still had problems with "swaps" from the old currency - so they were constantly reissuing new notes and new coins that needed to be exchanged - but many of the old notes or permission slips would have value on the black market
Another situation: in Brazil 90's, inflation was rampant and a new currency was being planned. But, until BRL was printed as the final currency, a few "forks"/new money were introduced, like URV (for one year), which set the market value for the new BRL, something like 2500:1 old-money:URV then 1:1 URV:BRL. So, the analogy: a fiat fork did happen and a two-currency system was put in play, until the "miners" decided to ultimately put the hash power into just one fork.
Typically people prefer to hold on to both coins after a split. So there's not actually that much sell pressure, meaning both coins can tend to exist together.
You can "cash out" whenever you like, obviously.