While democratic socialism can be traced back to the early 20th century, it would be a mistake I think to regard it as a fully-formed or cohesive ideology to which political groups did it did not subscribe. You can make an argument for viewing the Labour Party as being a democratic socialist organisation, but it's not a term that they would have necessarily drawn on to describe themselves. Complicating the matter here is that the Labour Party in this particular period is much less centralised than today - it's still meaningfully a federation of affiliated organisations and parties, with potentially significant differences in beliefs and policy ideas between groups.

In any case, being a Labour supporter was hardly unusual by the 1920s, with the Labour Party supplanting the Liberals as the progressive party of government over course of the decade, aided by the expansion of the franchise that enabled all adult males to vote. Support for Labour was naturally correlated with class (particularly unionised industries), so you could say that it could be meaningfully rebellious for someone from a more privileged background to be a forthright Labour supporter, though this was hardly unknown (and many early Labour MPs in this period came from quite respectable backgrounds). Affiliated movements like the Fabians were essentially middle-class socialists with a reformist bent, and were quite influential on the wider movement at times. However, to really shock the neighbours you'd be better off joining an explicitly revolutionary group, most likely the new Communist Party of Great Britain, formed in the Bolshevik model early in the 1920s.

If we had to point to one group that most resembled democratic socialists in this period though, the Independent Labour Party is the.obvious choice. The ILP was, during the 1920s at least, one of the affiliated organisations that made up the Labour Party, and one of the most influential to boot. At their peak, maybe a third of Labour MPs were affiliated to them, and in parts of Britain (such as Scotland) they basically were the Labour Party - local branch systems and membership were based almost entirely on ILP structures. They formed what might be seen as the first incarnation of an official 'Left' faction within the Labour Party, but unlike subsequent incarnations were significantly institutionally independent (as the name suggests) of the Labour Party itself. Their structures and ideals drew on older methods of working-class organisation and civil society (mutual aid, temperance, co-operation), resulting in a notably decentralised system with a large emphasis on internal democracy and branch autonomy. However, they had fewer concrete ties with the trade union movement, which by the end of the 1920s was becoming increasingly centralised and influential within the Labour Party.

Ideologically, the ILP adhered to a meaningfully revolutionary programme, both in terms of goals (public ownership, redistribution of wealth etc) but also in terms of speed - while they envisaged their goals being achieved peacefully through parliament rather than insurrection, these aims were explicitly near term ("in our time") rather than a vague, far-off goal. This put them at odds with other elements of the Labour Party, who argued for a more gradualist approach more in line with social democracy (though even this label doesn't sit completely comfortably with British politics). This divergence culminated in the ILP formally withdrawing from the Labour Party in 1932 rather than compromising their political vision, having already ran their own candidates in the 1931 General Election (returning 5 MPs). Thanks to the Second World War, the 4 ILP MPs remaining after the 1935 election remained in office until 1945, styling themselves as principled (but largely isolated and impotent) voices of opposition trying to chart a route between the more dynamic revolutionary vision of the Communist Party, and the increasingly bureaucratic and centralised Labour Party. Rebellious they certainly were, but the kind of rebels that the system could tolerate without too much fuss.