Mining pools, exchanges and processors have no direct say. All that is needed for the BIP to be merged is agreement among the lead developers. That would be Van der Laan, Gavin Andresen, Jeff Garzik, Gregory Maxwell and Pieter Wuille. Correct me if I missed a name or am in error.
Miners have most of the power. They can immediately protest and scare the developers into changing their proposal or adopting another. They also are the ones that have to ultimately accept the code so whatever the developers implement can be ignored or rejected.
The only power the developers have over the miners ultimately is the fact that they can walk away as volunteers and refuse to contribute if the miners don't accept their updates. This isn't of much consequence because other (likely less talented ) developers would fill the void. The miners, node operators, merchants and exchanges will likely accept the soft fork because they agree with it and trust the judgment of the developers.