In the first sentence, the lack of a plural could emphasise that many of the shipyard workers are likely to lose their own livelihood.
In the second, the plural could emphasise that many shipyard workers may lose all of their livelihoods, their collective livelihood, the one they share.
However, semantically, I don’t think it makes a difference, at least not in this case.
It has very little to do with grammar; I think it’s completely stylistic. All of the permutations make sense to me equally.
The falling orders for new ships mean that many shipyard workers are likely to lose their livelihood.
makes just as much sense as
The falling orders for new ships mean that many shipyard workers are likely to lose their livelihoods.
Likewise,
Many ship workers could lose their livelihoods because of falling orders for new ships.
seems to mean the same thing as
Many ship workers could lose their livelihood because of falling orders for new ships.
The plurals seems purely stylistic, because the emphasis that could be given by the plurals doesn’t seem to come into the text. Anyone who knows English will read them effectively the same way. Hope that helps!
Best Answer
Although there are many people that claim that "none" and "neither" should always take the singular form, it's always sounded odd to me, so I decided to dig into it a bit further.
For example, here is an Ngram of "none of us is" / "none of us are":
In this case, we can clearly see that "none of us are" - i.e. the supposedly "ungrammatical" form of the sentence was vastly more popular until roughly 1880. In 1880 or so "none of us is" began to take hold, and in the 1920s there was a steep decline in the use of "none of us are" - perhaps in response to overzealous copy editors enforcing the so called rule.
Most interesting is what's happened since the 1990s, where "none of us are" has shot back to prominence, leading to the fact that "none of us are" is now the dominant form again.
As suggested by snailplane, here's some interesting additional reading which seems to confirm my opinion that "none/neither should take the singular" is a wholly invented rule:
http://johnemcintyre.blogspot.co.uk/2010/01/with-malice-toward-none.html
http://motivatedgrammar.wordpress.com/2009/01/08/none-is-none-are-grammar-according-to-clarkson/
As a native speaker, I've always just gone with choosing the verb as if "none" or "neither" was not taking part in the verb choice.