What do you call the vein in your neck? the one that gorges blood out if you was to slice it like sweeney todd?

The jugular vein is the one close to the surface that you can see in people's necks; the carotid artery lies deeper but it is the one that pulses and you take a pulse from in a cardio workout class. However, your simile for water is not really very good: neither would "ooze speedily" and arteries do not ooze at all, and cutting someones neck would be like a fountain or a waterfall not a stream. IMO, I would look for another description of what you are trying for.