William Heinemann reprinted THE SUNBIRD several times: in 1972 (reprinted before publication), 1974, 1977, 1985, and 1988 for a Uniform Edition.
I found the signed copy below on eBay; it is not in my collection
Below: A copy that is owned by Nick De Carteret, who was able to meet Wilbur at the farewell party that Macmillan threw for Wilbur at Mossiman's in Belgravia at the time Wilbur was switching to Harper Collins as his publisher
I have owned and sold numerous signed copies of THE SUNBIRD. All below are first printings.
Doubleday reprinted its first edition so not all copies are first printings of the first edition.
The copyright page has to state FIRST EDITION IN THE UNITED STATES OF AMERICA
Many copies of this edition were remainder-sprayed by Doubleday. It is tough to find a copy in collectible condition that has not been remaindered. I have owned copies with remainder spray on the bottom page edges of the book. Below is an example of what remainder spray looks like on Doubleday copies of GOLD MINE, DIAMOND HUNTERS, and EAGLE IN THE SKY, with THE SUNBIRD in the middle without remainder spray.