A mining full node is a full node that not only holds a block chain, accepts other blocks from others according to its protocol rules, accepts broadcast transactions that come along but also constructs a new block from its mem pool on top of it according to his own rules, and broadcasts that new block.
Non-mining full nodes have quite a bit of power if they are operated by exchanges or payment processors. If you can't spend your coins in those contentious blocks they are pretty much useless.
The point is that
if these entities install nodes that don't see blocks, then they kill their own business. Concerning exchanges, what is their business ? Exchange coins. If two block chains exist, they have all the interest to list both of them, so that people can exchange one in the other. Exactly as on ethereum.
Can you imagine an exchange telling its customers "nope, sorry, we can't accept deposits or withdrawals, because we installed a node that refuses the block chain that miners are making" ? (this in the case where the miners didn't fork, but the "UASF" node is installed: it stops at the first non-segwit block, and comes to an eternal halt).
They are in the same catch-22 as the miners. If the split took place, they could consider installing such a node. As long as the split didn't take place, installing the node stalls their business.
And IF the miners did split, do you think an exchange is going to forego the huge benefit of having all bitcoin users spend one of their tokens (every bitcoin user now has TWO coins, one on the segwit chain, and one on the non-segwit chain, like with ETC/ETH) to buy the other one, and take fees on all these transactions ?
So they WILL list the old chain just as well as the new one.
Mind you that the minority chain will make blocks less fast. So the segwit chain will be more congested than the non-segwit chain (the segwit chain will be in minority - if it were in the majority, there wouldn't be any discussion: it is a soft fork, so it would IMPOSE its majority on the entire chain). It will be even more of a pain to transact on than the other chain. Suppose now that users prefer the classical bitcoin chain in the market. If the price plummets, miners will switch back to the old chain, and the segwit chain will maybe die (due to bitcoin's slow difficulty adjustment). All segwit chain transactions lost ! If you sold your classical bitcoins, they're gone !
The risk also exists in the other direction. If segwit becomes majority, and the classical chain is overtaken, then it will simply be orphaned (due to the soft fork nature). Everything you did on the classical chain is forgotten. If you bought classical coins and sold your segwit coins, everything is gone !
So you better think what to do, because you have to buy the chain of which you think it is going to win ; but buying the minority chain is going to be difficult (it will be VERY SLOW) ; however, the majority chain may disappear if the minority (segwit) chain becomes majority. What are you going to do ?
(I would cash out....)