What if after the world war II the United States had become a social democracy ,like GB ?
Better unions might be part of the answer. I think the US unions worked with the organised crime syndicats (Hoffa etc.) and strong unions are part of the equation. (I'm not sure about nationalized industry. It is something that changes over time in social democratic thinking.)
Unions aren't just a part of a political movement. They are also a alternative to political action through the goverment at one level or another. For example, I like to point out that Sweden has no minimum wage.
Unions work like guys with hammers. They see all problems as nail problems. Unions see problems as unions as the solution. They would also prevent the tipping over to some worse form of socialism.
Indeed they did, our early socialdemocrats were pragmatists so they made a deal with the unions and the industrialists called Saltsjöbadsavtalet. In this deal the unions said we will not strike if our members get decent wages and benefits, the socialdemocrats said we will not nationalize and finally the industralists said ok we will give decent wages and benefits but no strikes or other sabotage. The industrialists in that time not only looked at profitYes, I think you definitely need stronger unions. Not just stronger unions but also a more centralised form of trade unionism. Many unions in the US tended toward a sectional craft unionism sort of organisation(many small unions). With the exception of the UK, most nations with large social democratic parties have a more system-wide industrial unionism (a small number of large unions).
Yes, it is true that support for nationalisation is something that varies in different national traditions in politics. For instance in the UK and in Australia, traditionally social democrats strongly supported nationalisation. In Sweden they opposed it. It seems that in nations with highly conflictual relations between capital and labour (UK and Australia) nationalisation was supported, whereas it wasn't in consensual nations (Sweden).
The US has more the pattern of conflict seen in UK and Australia, so I can see at least for a while social democrats supporting some forms of nationalisation.