Interestingly I am also British and my impression from American friends and acquaintances was generally the opposite to yours: many of them had volunteered in the past or were involved with volunteering, which is something I came across less frequently in the UK (I emigrated 4 years ago so this may have changed, I suppose). This impression is borne out by the report above.
While the social issues faced in every country differ greatly for any number of reasons governed by history, demographics, etc. etc., there are enough similarities between the two countries as to make you sound very insensitive to the issues faced by many inhabitants of the UK.
To use one small example, the Gini co-efficient measure of income inequality (after taxes) for the UK is closer to that of the US than it is for almost every other European country . I don't think anybody would dispute that there are a lot of poor and disadvantaged people in the UK.
The main difference between the two countries has less to do with the willingness of citizens to donate time or money to charitable causes than it does to do with the "social safety net", provided by the government. That's a whole different subject though.
Britain's Gini coefficient is substantially distorted because of London. The Gini coefficient is a pure measure of income inequality, so is massively influenced by the presence of people with exceptionally high incomes. It tells you almost nothing about the gap between the middle class and the working class in a developed country. When your country includes a megalopolis that's the world's most popular playground for billionaires and tax haven for multinationals, that distorts the figures somewhat. Factor out a couple of London boroughs and our Gini index changes dramatically. Likewise, France's Gini index would plummet if they had a territorial claim to Monaco.