The speed comes from the fact that private and consortium blockchains tend not use Proof of Work (at least not as the only algorithm; you could mix and match, if you wanted to). Instead, such blockchains tend to use voting-based protocols (e.g. Proof of Stake), where a certain subset of nodes are trusted with voting rights, who together vote to approve the next block. In some systems you also have automatic punishments built in to incentivise correct behaviour (i.e., if someone attempts to fork a blockchain --- for example by voting for 2 conflicting forks -- and if other nodes see any evidence of that, the node who attempted the fork could automatically lose their rights as a trusted validator, and may also automatically lose a lot of money or face legal fees.)