We cannot date your Prayer Book, but if the name of the University Printer is given on the publication details page at the front, his term of office will enable the Bible to be dated within a range of years.
Another clue can be found in prayers for the monarch and in the State Prayers at the end of the Morning and Evening Prayer services, where the royal personages vary over time. The Society of Archbishop Justus has a comprehensive list of changes to the text. See also David N Griffiths', The Bibliography of the Book of Common Prayer 1549-1999, British Library, 2002, ISBN 0-7123-4772-0.