The singular here, but not necessarily in all uses of terms covering groups.
It should be noted that this differs from the similar case of an action by a group that could be reasonably be treated as either singular or plural (though with different preferences found more often in different dialects). This would cover the use of a country's name to represent its official policy or government's position, but not the use here.
This use is treating a place figuratively in talking about its people, and so that single noun is the figurative subject, and can only be singular.
Such figurative use becomes split again though, if the place name itself is plural; e.g. The United States, The Bahamas in which it can be treated as a single entity or as several.
Such figurative use is also rather old-fashioned, these days. It would be more usual to just use something like the more literal "the people of York speak English", or the more precise "the vast majority of people in York speak English".
Now, if the entire phrase is to be the name of a group in itself (adding, due to your comment), this is slightly different again.
If it was for me to choose between naming a group "York Speaks English" and "York Speak English", I would favour the former for three reasons:
- It's a use of the figurative use described above, or something close to it.
- When it comes to the freer cases, the singular is more common in British use, so I would favour it for York.
- I can right now think of at least one organisation with "Speaks" in its name in this way, but none with "Speak".
However, I'd also have some hesitation with the name as a whole. Three of us here at first reading missed quite what you meant with "The name of group can be" in your question as first written. That in itself suggests that we will not be the last. Verbs in names of group are more common now that even just a short time ago, but are still relatively unusual. I think I would take a different tact entirely. I must say though, actually giving something, whether a group, a product, a work of art or whatever, a name must be one of the hardest tasks we ever put our English skills to!