You'll still need access to the blockchain, either directly (to a local node) or using a remote API like blockchain.info etc.

A private key will allow you to sign the tx inputs, but you'll still need to get those specific outputs from the blockchain to use as inputs in the transaction you are planning to create and transmit. This can be done quite quickly (milliseconds): For example even the Blockchain Android/IOS app creates tx's using the remote API to the blockchain very quickly (a few seconds).