I'll try to answer as best I can, because I notice that questions about Methodism seem to get very few answers, but this is somewhat outside my area and I don't have as many books to hand as I would like. I will talk mainly about the Labour Party, but it's really important to place this in the context of a wider debate around Methodism and the labour movement which stretches back into the nineteenth century. As another recent questioner asked, an important argument - stated perhaps most famously in EP Thompson's The Making of the English Working Class - is about the idea that Methodism was 'a politically regressive, or "stabilizing" influence' and 'indirectly responsible for a growth in the self-confidence and capacity for organization of working people' (p.45). This isn't an argument I am qualified to go into, but it informs a lot of what is being insinuated when someone contrasts 'Methodism' and 'Marxism'. As you point out, the connection to trade unionism is more obvious: the Labour Party's immediate predecessor, the Labour Representation Committee, was formed chiefly by trade unions in order to secure parliamentary representation in the aftermath of a series of disastrous judgements - but these trade unions, as Thompson and others suggested, were themselves influenced in one way or another by Methodism (and perhaps more broadly Nonconformity).

The quote itself is associated with Morgan Phillips, general secretary in the 1950s. I suspect the person he was thinking of most strongly was Keir Hardie, the founder of the Independent Labour Party in 1893, who was a lay preacher as well as a trade union organiser. His biographer, Kenneth Morgan, makes a similar claim:

'I found him a man who was not only an idealistic crusader, but a pragmatist, anxious to work with radical Liberals whose ideology he largely shared, subtle in building up the Labour alliance with the trade unions and the other socialist bodies, and supremely flexible in his political philosophy, a very generalised socialism based on a secularised Christianity rather than Marxism. 'Socialists,' he proclaimed, 'made war on a system not a class'....He was no economist and was ill-informed on many issues, but he had uniquely the charisma and vision that any radical movement needs' (pp.89-90).

Christianity was important to a lot of early Labour Party members and leaders, and particularly the non-established churches. Another famous quote, after all, describes the Church of England as 'the Conservative Party at prayer'! - although they did secure some support from clergymen and even bishops by the end of the 1920s, most famously Conrad Noel the (Anglican) 'Red Vicar' of Thaxted. Nye Bevan, too, came from a Nonconformist family - his mother was a Methodist, his father a Baptist. However, the association between Methodism specifically and the Labour Party can also be overstated. After all, a big part of Labour's support in areas like Liverpool or Glasgow came not from Methodism but from the Irish Catholic community. Ramsay MacDonald, whose outlook dominated the Party until the controversial 1929-31 government, went to a Presbyterian school. Some important figures, like Clement Attlee, seem to not have felt strongly religious at all.

As I've suggested, the main point of a quote like this is not to emphasise the importance of Methodism but the relatively lack of importance of Marxism - the alliteration is probably a factor in the disregard for trade unions which you note! I will comment just briefly on this history. The largest pre-WW1 Marxist organisation in Britain was the Social Democratic Federation, which did take part in the formation of the Labour Representation Committee and later became the British Socialist Party - admitted to the Labour Party in 1916. Unlike in other countries in Europe, where socialist parties often preceded the creation of trade unions, in Britain the process was the reverse - which led to a correspondingly smaller Marxist movement and a less prominent position in the party compared to the (predominantly Liberal-leaning) trade unions. By the end of WW1, Marxists in Britain were increasingly inspired by the example of the Bolsheviks in Russia - and by 1921 much of the British Socialist Party had merged with other smaller groups into the Communist Party of Great Britain, a member of the Third International. Despite Lenin's calls in Left-Wing Communism for British communists to seek affiliation to the Labour Party, this was repeatedly rejected - although many Communists did join as individual members, and two (Walton Newbold and Shapurji Saklatvala) were elected as MPs, with Saklatvala elected as a Labour MP. In 1930, the Labour Party introduced a list of proscribed organisations including the CPGB and hence the influence of 'official Communism' became increasingly limited.

However, that isn't quite the whole story. As Tony Benn stated in Arguments for Socialism, 'never since the earliest days of the Labour Movement has Marxism itself been regarded as a disqualification for party membership' (p.34). Several Marxist organisations, including perhaps most famously the Revolutionary Socialist League (Militant Tendency), operated within or dissolved themselves into the Labour Party despite the proscriptions. Many Labour Party members had friends or family who were openly Marxist, or themselves had been former members of the CPGB - Nye Bevan, for example, read Marx at college and would refer to him in speeches. Bevan is a useful reminder that Marxism and Methodism (as well as other traditions like Fabianism) were not necessarily in opposition in the way that quote implies. Ralph Miliband noted with regard to the 1931 crisis that:

'Some of the members of the Executive had long been Marxists, or at least marxisant. But there were others who had, until 1931, dwelt fairly comfortably within different traditions, either Christian Socialist or Secular Fabian. Some of these now came to question their earlier assumptions concerning the inevitability of parliamentary gradualism.' (p.197)

Two points can be seen here: the presence of Marxists or fellow travellers at senior levels of the Labour Party, and the fact that discontent did not necessarily align neatly along Marxist/Methodist lines.

Similarly, Harold Laski authored a pamphlet called Marx and Today, making the case that:

'In being wise and courageous about our own problems, we can help this sorely-sicken Europe to be wise and courageous about its own problems too. But the supreme need is for us to face them in the knowledge that only upon a Marxist basis can we diagnose them correctly, and hope, thereby, successfully to solve them' (p.25).

Doubtless Laski was a marginal figure and an intellectual, but he did carry some weight within the Party and his pamphlet would have been read and probably agreed with by some Party members, especially in the immediate aftermath of Soviet victory on the Eastern Front.

A few final remarks, however, for fear of insinuating that Marxism was only influential on a general marginal Labour Left. Peter Mandelson, as a teenager, was a member of the Young Communist League. Marxism Today, the journal of the CPGB, in the 1980s took a shift which arguably bringing them to the right of the Labour Party. The so-called 'New Times' thesis, associated with figures like Eric Hobsbawm and Stuart Hall, provided an intellectual basis for New Labour even if its advocates likely wouldn't have recognised themselves in Blair. You can read some more about Marxism Today here: https://www.theguardian.com/politics/2015/sep/29/marxism-today-forgotten-visionaries-whose-ideas-could-save-labour. Certainly, I don't think that had much to do with Methodism!

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