It’s not mandatory to have your hair and makeup trial on the same day as a dress fitting but some brides choose to. Personally, I didn’t as I wanted to see everything come together for the first time on the day.
If you’re a worrier, then I would say booking your trials on the day of your final fitting is a great idea. It might help settle any nerves you have about how your hair and makeup will work with your dress. If you do intend to book everything on the same day then I would suggest booking an afternoon dress fitting so that you can have your hair and makeup done in the morning. Bear in mind, most stylists have crazy diaries from April - September with most mornings already booked up for weddings. You may have to book well in advance or be fairly flexible to make this work.
If you’re pretty chilled and are confident with your choices then I would say do them independently, it will give you another thing to look forward to on the day!