Russell argued for a "scientific society"
Science at the beginning of the 20th C was still a gentlemanly pursuit. Now it is heavily professionalised and employs - directly and indirectly - many tens of millions of people. More importantly, a scientific society is akin to a rationally run society and this is the essential principle that forms the background to Enlightment notions of a flourishing society.
For example, secularism in Western societies was a rationally introduced principle that arbitrated between various religious denominations by being a common neutral point. (This will probably require renegotiation in the future given the remarkable growth of athiesm over the 20th C as secularism allies itself more naturally with the athiestic view by the very virtue of its neutrality).
where war would be abolished
War before the 20th C was naturally limited by virtue of the actual technology available to contending parties and nations. War had its casualties but the number of casualties were limited. But given the immense increase in the power and the reach of military technology then it makes sense to call for the abolishing of war. This is not a consequence of a scientific society per se, but again a society run on the principles of reason and flourishing.
the growth of population limited
Worries about a demographic explosion were a common worry at the beginning of the 20th C with a corresponding increase in the interest in eugenics with tragic consequences such as forced sterilisation in Indira Gandhi's India and the holocaust in Western Europe under the instigation and leadership of Nazi-run Germany of Jews, the Roma and the mentally ill and physically disabled.
However food production under scientific principles was hugely increased. The problem then became that of effective distribution.
and prosperity shared.
In the aftermath of the Second World War social democracy in Western Europe was an effective mechanism of equitably distributing the goods produced by society. Starting in the eighties the neoliberal revolution inverted this and inequality began to build up again where now out of the top one hundred economies in the world, sixty nine of them are corporations. Inequality is recognised as huge global problem. There are useful suggestions to rectify this, for example, Stiglitz has suggested a minimum of 20% corporation tax to be globally enforced which avoids their arbitraging of taxation jurisdictions by transnationals, or the Tobin tax on financial transactions.
He suggested the establishment of a "single supreme world government" able to enforce peace
This was urged by Roosevelt and which led to the creation of the League of Nations which did not work well and was short-lived; a second attempt resulted in the creation of the UN; globally, there is a whole ecology of such global institutions. Others include the WTO, the World Bank and many NGOs such as Amnesty International or the Red Cross.
claiming that "the only thing that will redeem mankind is co-operation"
This is a very old idea, rather than a new idea; it has always been cooperation that redeemed mankind from its atavasitic and competitive instincts, which in its bare form leads to war; and in this modern age, cyber warfare which takes two main forms - technological - for example actually hacking into networks - or psychological ie cyber-bullying and psychological manipulation.
Religion for example is derived etymologically from a Latino word meaning 'come together' and which is reflected in the notion of the congregation or mass in Christianity, or jumma in Islam which has the same connotation as congregation.
I haven't read this particular book by Russell so I can't comment on your final question; however, Hannah Arendt in her book, The Human Condition does cover similar grounds where she observed drily that scientists did not refuse to develop atomic weaponry but having done so were surprised 'that they were the last to be consulted in their use.'
This points out that the power released by scientific technique in society remains in the hands of those who wield power - aka the establishment in all it's various forms, national and internationally - and not exclusively that of the scientific establishment, though of course they have some relation to political power, both legislative, executive and judicial.