The context of the song being sung is that the conscripts have been invited to a Christmas party with the officers, and folk music is brought up with the implication that the Officers, as educated people, are familiar with their country's history and folk music, while the conscripts would only be familiar with popular contemporary music. The conscripts sing with increased aggression because they are angry at being dismissed by the officers as uneducated louts and also, as you note, there's an undercurrent that this song, associated with the Peasant's Revolt, also reminds the officers that the conscripts, common people, outnumber them, and therefore should not be treated so poorly as they might rise up in response.
From Only in the Common People: The Aesthetics of Class in Post-War Britain by Paul Leslie Long
Wesker explored the value of folk in his most successful play, Chips With Everything which, along with the festivals, opened in 1962. This play follows a group of RAF conscripts through their national service training. Drawing upon personal experience, Wesker uses the tensions of this milieu to examine the enduring British class system and some contemporary cultural ideas and conflicts. Towards the end of the first act the men attend a Christmas party laid on for them by the officers. The odious Wing Commander reveals his estimation of them when he announces that in return he had expected ‘A dirty recitation, or a pop song. I’m sure there’s a wealth of native talent among you’ (Act 2, Scene 7). Pip Thompson, an upper-class recruit ‘slumming’ among the ranks, realises that the festivities are merely a cynical experiment. The representatives of the establishment seek confirmation of the docility and baseness of their subordinates. A surprise is in store however as Thompson encourages one soldier in a recitation of a dialect poem, and a minor battle of wills ensues. In turn, the officer encourages abandoned revelry and laughter, to the accompaniment of a rock`n’roll band engaged for the evening. In response, Pip whispers into the ear of the guitarist who begins to play an arrangement of ‘The Cutty Wren’. Quite spontaneously, the men join in the singing of this ‘old peasant revolt song’, which is delivered in a rather menacing manner, before the curtain falls. This vignette neatly delineates the politics of taste and class that were at issue. The idea of a shoddy mass culture is there, with disdainful reference made to Elvis and pop music in general. In opposition is the sense of an ‘authentic’ culture, which threatens by its very nature to galvanise the men into action. By fault or design there is an implication that the knowledge and understanding of this folk music is an immanent quality of the men, individual and collective. In this same vein, Jazz was approved of and included in festivals as: 'the last of the authentic folk music’. One contemporary writer considered it to be a bridge between high art and the popular culture of the people, underlining its authenticity