Kentish cobnuts are usually eaten when they are green and fresh; however they are also eaten later in the season when they become sweeter and the kernels begin to dry. Kentish cobnuts are available from late August-December depending on the season. The peak is usually late September/early October. (Roughway Farm, Kent) Similarly, wet walnuts from Kent can now be found in greengrocers' shops. So I doubt that they were gathering nuts in May.

Common Hawthorn, also known as May Blossom, Crataegus monogyna, may tree, quickthorn. Flowering in the middle of May, its folk name of may blossom derives from the older calendar when 1st May, or May Day, coincided with its flowering. (Plewsgardendesign)

The annual Garland Day in Lewes took place on Bank Holiday Monday. The colourful event is organised by women’s morris group The Knots of May, named from the tradition of gathering knots or garlands of flowers to mark the arrival of spring. So the custom is to pick knots of may or knots in May.