For the first time in Bitcoin history, if miners withdraw their mining power from mining BTC, transactions on the BTC chain will grind to a halt. However, Bitcoin Cash can survive any drop in mining rate.
Bitcoin Cash was built so that if it starts to take too long to mine blocks, it has an Emergency Difficulty Adjustment(EDA) to make it easier to mine blocks. For the first two weeks since the fork, there have been a few EDAs and it has dropped the Bitcoin Cash difficulty to 13% of the legacy Bitcoin's chain.
Bitcoin Cash was able to add this EDA code through a contentious hard fork - something that is really only possible with a large enough community coming together to update to new clients, get miners to mine it at a loss, exchanges to list it, and merchants to accept it. No easy task, but there were enough people fed up with Blockstream's takeover of Bitcoin Core development to make it happen. Legacy Bitcoin does not have this safety feature and cannot have one unless a hardfork is performed to include it, and with the current landscape that is not at all likely.
Up until now, there was no incentive for miners to leave the Bitcoin legacy chain. ASIC miners built specifically for Bitcoin's SHA256 mining algorithm needed to mine Bitcoin to be the most profitable, but now that can change with Bitcoin Cash. What happens if it becomes more profitable to mine Bitcoin Cash, and miners quit mining legacy bitcoin and make the switch to Bitcoin Cash?
A Chain Death Spiral occurs.
Less mining on the legacy Bitcoin chain will lower the hashrate, but it won't be able to adjust the difficulty down, causing miners to leave because it isn't as profitable. As more miners leave the problem gets worse and a feedback loop results in the dreaded Chain Death Spiral. The slower blocks are found the more miners leave until its impossible to get a transaction into a block at all. Even though Bitcoin has the default difficulty adjustment every 2016 blocks, without enough hashpower, it can never get through those blocks to reset it. All your BTC would become worthless overnight.
Bitcoin Cash to the Rescue!
Luckily, anyone who hasn't been paying attention will be fine in this event. They will have an equal amount of Bitcoin Cash that will takeover as THE "Bitcoin" since it will become the longest fork of the original chain. The value of legacy bitcoin will transfer over (more or less) to Bitcoin Cash, so its fiat value will remain relatively the same as well.
The biggest losers in the event of a Bitcoin Chain Death Spiral would be those who sold their Bitcoin Cash for Bitcoin after the fork - they would be left with nothing.
The biggest winners would be those who traded in their Bitcoin for Bitcoin Cash, especially if they did it at a 1:10 ratio. ;)