I sent emails to the supermarkets recently asking them about their beef and whether it was grass fed or grain fed. This is the response I got from Waitrose:

Cattle are finished all year round to maintain a regular and consistent supply therefore during the summer these cattle would be finished from 100% forage diet and our standards insist that forage must be at least 75% of the diet (winter). Forage comes in several guises, the common ones being grass, hay, silage but also whole crop cereals, maize silage's, roots and other fodder crops. Farmers grow and conserve these crops as part of their farm rotation in order to finish these cattle. Therefore there is a good nutrient supply that supports the diet even during winter months. Grain my be used for feed as long as the requirement for 75% forage is maintained.

Like beef, lamb is predominantly forage/ grass fed, but may receive supplemented feed where forage is short or unavailable.
So potentially, depending on the time of year or the weather over winter all of their own label beef is grass fed including their essentials range.

I haven't had any responses from the other supermarkets yet.