The easy answer is to capture a city with a new religion from one of your enemies. So you're forced to get a fast Astronomy and your progress is stalled until then. Maximize the benefit from the war by trying to capture a city that has more than one religion that you don't. Switch to organized religion to get out the Missionaries you need to send home. If you're a peacenik at heart then gift back the city once you've transferred the religion(s) back to your home continent.
The hard answer is less sure, but it also relies on Astronomy to get contact with other civs. Activate open borders with everybody you meet, each turn there will be a chance that a religion will spread to one of your cities. However, the chance of religions spreading is based on the amount of religions currently in a city. So a mistake you would probably make is to spread the one or two religions you do have to all of your cities. For this trick to work you want to have as many cities with no religion at all in them. If you're really desperate try to produce Great Prophets to gift them to the AIs with holy cities to encourage them to build the Shrines. Since you're across the water they are very unlikely to be sending you Missionaries, so a Shrine + Open Borders + No Religion is the best odds you can get of a religion spreading on its own.