Early May Bank Holiday

Museums should be open, maybe some private ones shut down, but generally things like museums, holiday spots, tourist attractions are busier on a Bank Holiday than other days.

The businesses that will be closed will be banks, government offices, corporations, etc. Pubs, restaurants, most retail, things that you are interested in as a tourist, will be open.

Edit: Lodging can also be an issue if you were to be planning to be in Cornwall, near the coast, or other popular tourist areas. They fill up fast and will charge a premium price.

Gotcha. So basically just like say Labor Day here.

Yes, it is essentially like a Monday Holiday here, same concept, a public holiday. The "Bank Holiday" name really does not resonate well with Americans.

Local buses [aka City buses] will run to a Sunday schedule on the Monday, although long distance ones [coaches] are on a normal schedule. Outside London and it's commuter routes trains generally run to a normal schedule. However, like on all Bank Holidays this is a favourite time to do railway engineering work- so be prepared for diversions and bus replacements on the trains. This is way too far out to know if that will be happening (8 to 12 weeks is the normal planning time for that work).
Museums in York will be open, as will all food stores.
The early may Bank Holiday [whether Americans like that phrase is immaterial, it is what it is called] stems from May Day festivities for the beginning of Summer- dating back to Roman times. Very often you will find special events happening in connection with that theme.

You'll be fine. Offices, schools and banks will be closed. Musuems, shops, restaurants will operate. (You might find one or two restaurants or shops closed be they're the exception to the rule.)

However, check opening hours as they might be different than regular days.

Transportation will run. Again, check the schedules for any changes.