All very good answers. The trick is figuring out how to make the bike handle well on the track. Generally, the 18" tires present a good handling and tire (compound and low profile) option. But, you need to make mods for ground clearance. If you are working with rules, they will limit your options. If you are not working with rules, your options can be anything you wish.
It's expensive to make a Vincent handle at the top of its ability, which is why you will get peppered with questions about your ultimate goals for the bike.
Often it is not clear what compounds and tires are available. Generally you are looking for information like this:
You will also need some info on compounds.
When you see the term "Cantilever", as in the AM23 selection, it means the tire has been designed to be used on a narrower rim than is normal. As Glen mentions, you have to factor in the recommended rim sizes, which tend to be on another chart.
Often it is a matter of finding a tire guy or a tire shop that will give you the info you are looking for and supply you with "fresh" tires, not just the tires in stock. The freshest tires will perform better. Once you feel you know what tires you have available for purchase it may slot you into a decision.
David