Very little. Churchill lost his post as Prime Minister almost immediately after the end of the war in Europe. His right-wing Conservative party lost by a huge margin to Clement Attlee's left-wing Labour party; by the time of the surrender of Japan and the Potsdam conference, he was already being replaced.

For background; it's worth noting that while Churchill had been Prime Minister, the Government had been a coalition, with Attlee as Deputy Prime Minister and other Labour figures in key positions. This had a couple of benefits for Labour; the nature of warfare necessitated state control of industry, and it allowed Labour figures to show their talents as a government.

In 1941 the Beveridge report had codified demands for state intervention in poverty, and while a victorious post-war Churchill would never have gone as far as Labour did, the consensus was that something would have to change;

Churchill...offered the prospect of...'national compulsory insurance for all classes for all purposes from the cradle to the grave', the abolition of unemployment...by government action to 'exercise a balancing influence upon development which can be turned on or off as circumstances require' and even 'a broadening field for state ownership and enterprise'

(Welfare States and Societies in the Making, Deakin and Finer)

It's also worth noting that Churchill was a former Liberal and would today be identified as a 'One-Nation' Conservative. He had been involved in post-WWI welfare measures including the 1911 National Insurance act and the Widows, Orphans and Old Age Pensions Act in 1925. That's not to say that he was a leftie at heart, just that he was not inherently against state intervention in the economy and support of the disadvantaged.

So while, to dip into counterfactuals, he almost certainly would not have gone as far as Attlee did in his changes, he recognised that Welfare reform was a necessity and would have offered some concessions of his own. This post-war period marks the beginning of a period labelled the 'Post-war consensus' which would last roughly thirty years and see the British economy governed on vaguely Keynesian (to be simplistic; tax and spend) policies.

Back to the 1940s; for the Welfare reforms passed by Labour in the mid-late 40s, Conservative support was not required due to the stonking majority Labour had won, and the Tory party did vote against, for example, the NHS (annoyingly I couldn't find the voting totals anywhere; if I dig them up I'll put them in). By the time the Tories were back in power in 1951, there was no appetite to dismantle the extremely popular reforms. The Conservative government had a tiny margin (it had in fact lost the popular vote by more than a million votes, but won 12 more seats due to the vagaries of First Past The Post).

Churchill characterised the situation as "Four-fifths of both parties agree on four-fifths of what should be done, and after all, we all sink or swim together on our perilous voyage ever accelerating into the unknown.'" Immediately before this, the Labour government had had to roll back some of their health reforms due to the cost, but the conservative manifesto of 1951 makes no committment to scrap their reforms;

Here is the surest way to keep our people fully employed, to halt the rising cost of living, and to preserve our social services. Hard work, good management, thrift - all must receive their due incentive and reward.

Housing is the first of the social services. It is also one of the keys to increased productivity. Work, family life, health and education are all undermined by overcrowded homes. Therefore a Conservative and Unionist Government will give housing a priority second only to national defence. Our target remains 300,000 houses a year. There should be no reduction in the number of houses and flats built to let but more freedom must be given to the private builder. In a property-owning democracy, the more people who own their homes the better.

In Education and in Health some of the most crying needs are not being met. For the money now being spent we will provide better services and so fulfil the high hopes we all held when we planned the improvements during the war.

We shall review the position of pensioners, including war pensioners, and see that the hardest needs are met first. The care and comfort of the elderly is a sacred trust.

So to refer back to the original question;

How much does Winston Churchill actually contribute to the Post-War Welfare Reforms in Britain?

Almost nothing directly, although he had a hand in laying some of the foundations they were built on.