Hi Rachel,
I have loads of St. Mary Bourne Bright family in my tree, and they have no gravestones either. I think they were too poor. There is a disused old Methodist chapel on the Stoke Road, which has a few headstones, but again I'm sure they had to be well off.
I know of one person by the surname Ralph living in Whitchurch, Hants, which is only about 3 miles from St. Mary Bourne.
There was also a Salvation Army building in St Mary Bourne way back, and probably a Baptist church, as there is a road called Baptist Hill in the village, but I expect neither of these would have burial grounds.
Regards,
Hampshire Grandma