Internet is definitely the most popular way Bitcoin transactions are broadcast across the blockchain network. Without it, thousands of computers around the world will not be able to validate the transaction to update the network. This will absolutly slow down the process of the transaction.
However, in case of total failure of internet, there are other alternatives to send bitcoin transactions. In fact in 2017, Blockstream, a company dedicated to improving Bitcoin functionality, has announced that they succeded to develop a method to send bitcoin via satellite without the need to have access to internet.
Since bitcoin transactions are completely public, data can be sent and exchanged via other methods then the internet. However, due to it's efficiency, internet will remain the main way for bitcoin transactions.