Bedfordshire Cap badge, Two Royal Engineers buttons, one normal size, one very small and one REME button. 4 items in total.
The Cap badge is missing its clasps on back.  I am afraid I do not know any of the history of them as they were in a small box of assorted buttons and badges from my parents.

The small Royal Engineers button measures 1.5cm in dia. Other is 2.5cm. in dia.
REME button is 2.5cm in dia.