The company now known as Royal Winton was founded in 1885 by the Grimwade brothers. Originally called Grimwade, the company became Royal Winton in 1930. Royal Winton has produced fine bone china and tea sets in lustreware, Art Deco, has made souvenir china and is most famous for it chintz products. Chintz is based on old calico patterns and features floral themes that cover the entire item. They are very pretty and colorful. The chintz patterns were very popular in the 1930s and were reproduced in the 1990s. Reproductions of the 1990s show the words "limited edition" on the bottom.
A miniature cup and saucer is called demitasse and is used for serving Turkish coffee or espresso.
Before you can learn the value of your demitasse cup and saucer, you need to identify the pattern. Some patterns are more in demand than others. Demand also varies on location.
If your demitasse is in the chintz group of patterns, you can find the specific pattern in a book:
"Collecting Royal Winton Chintz" by Muriel M. Miller.
You may be able to find your pattern on Google images or Pinterest. Once you know your pattern, you can shop around online to see the kinds of prices being asked. Asking prices, however, can be unreasonable so it's best to look at the amount the item was sold for.
You can also find your pattern (whether it is chintz or another motif) online at Kovels or Replacements.