No is the short answer. There are huge problems with your assertion that pre-Blair the Labour party was officially socialist. The main being that Blair's change to Labour's constitution was the first time the party was officially known as 'Democratic socialist,' previously it had simply upheld a commitment to nationalisation. Which is basically what clause IV was.

The problem with thinking pre-New Labour, Labour was a socialist party is nothing backs that up. Their postwar nationalisations were not done to out of any sense of giving workers more control, in fact many left the same managers in place as pre-nationalisation.

Postwar Labour followed an avowedly liberal programme, devised by Beveridge and Keynes, for social and economic programmes respectively. Both of whom were Liberal party peers.

As for harming relations with the US, not really. I seem to recall Truman wasn't a big fan of Attlee or the Labour party. Labour actually sent Keynes in a delegation to secure the Anglo-American loan after the war. The Labour leadership wanted 0% interest on a $5bn loan but ended up settling for 2% on a $3.75bn loan. The US didn't know that Britain would've taken anything it could've got as it was in far worse financial straits than many in the US administration thought.

They were still allies to the US against the Soviet Union, that much was never in doubt, both nations fought alongside each other in Korea. The only times the US would've found it hard to work with Labour, they were in opposition (79-97, though I'd argue '87 Labour and onwards would've been a stable force in government as Kinnock had successfully seen off the hard left by that point).

There are a lot of assumptions about "Old" Labour, the idea of them being socialist by any definition is perhaps the worst.

Figures of the party's hard left(formed when the left split in 81 over Tony Benn's deputy leadership challenge) have historically pushed this idea, from Tony Benn claiming the only two socialist Labour manifestos were 1945 and 1983 and Dennis Skinner more recently saying "A Jeremy Corbyn led government would be like 1945 all over again."

It's a reading of the Labour party's history that is intended to present Blair, Brown and Miliband as an abberation from 'Old' Labour, not something which actually seeks to understand why New Labour won and what changes were actually made to the party.

There is no evidence to suggest Labour followed a socialist programme by any definition at any point they were in government. Clause IV both past and present is purposefully vague, I'd say Blair's biggest mistake around it was including the phrase "Democratic socialist party," when referring to Labour, if you listen to any Gordon Brown speech you'll realise he uses the term synonymously with socialism and social democracy.

For him and New Labour, it simply meant 'centre-left.' It was a mistake because it did not lay out the fact that Labour is and always has been a broad church of liberals, social conservatives and socialists.

New Labour was either arguably the re-emergence of the party's social democratic tradition which left to form the SDP in the early 80s or the modernisation and amalgamation of 'old right' and 'soft left' ideas by the three sharpest political operators within Labour.

It's disingenuous to present them as separate from 'Old' Labour because the only Labour that had existed for the 18 years prior to 1997 had lost every election they'd fought.

So measuring the differences between a party in opposition mercilessly fighting itself, and a party in government which won three elections consecutively and became the defining force of a generation of British politics isn't going to yield results of any use in measuring which was more socialist. Neither 'Old' or New Labour was socialist by any definition.

As for the cold war, it's also worth noting Labour party figures were amongst those who founded NATO. Anti-Communism is a tradition that stretches back to the foundation of Labour when they decided to pursue the parliamentary road to socialism (again this doesn't mean they were a socialist party by any reasonable definition).

The Labour party's purpose is to represent the trade unions in parliament. That's Clause I of the constitution, this put them in diametric opposition with the Soviet Union, which prohibited independent trade unions. While there are many other reasons why Labour considered the USSR their enemies, in ideological terms this is the most easily explained. There are exceptions to this of course, Stafford Cripps being the main one in the 20s and 30s.

Tony Benn said of the Labour party, "It has never been a socialist party but has always had socialists in it." While he was saying this to promote socialism within the party, his point stands.

Sources:

Labour in Power - Kenneth O'Morgan

The Five Giants, a biography of the welfare state - Nick Timmins

Derek Fraser - Evolution of the welfare state

The End of the Party - Andrew Rawnsley

The Wilderness Years - dir. Michael Cockerell

A History of the Labour Party from 1914 G.D.H Cole

British Socialism: The Grand Tour(BBC Radio 4) - Anne McElvoy