Joan Begbie wrote Freelance the Pony. Freelance is the daughter of the New Forest Kitty, fastest pony in the New Forest. Freelance has many ups and downs before she is finally reunited with Judith, who loved her first and best. As far as I know, Joan Begbie wrote no other pony books, though she did illustrate Jean Cree’s A Pocketful of Ponies (Witherby, 1956).
Finding the book
Freelance can be pricey, but reasonably priced copies with dustjackets can still be found.