More freedom of action from within the EU, practically rather than in some abstract notion that 'well we could do this, but we won't because various powerful interests would make life very difficult for us if we did'. (And what are bilateral trade deals, those things that nobody gave a fuck about but that are now all the rage? They are a series of agreements between nations in which they agree to certain sets of rules. A trade deal with the US, for instance, would necessarily involve giving US businesses additional legal rights: rights to buy up UK interests and to have those investments protected. Our hypothetical l/w govt, by the time it gets into power, might in such a situation be up against US investors, backed by a US trade deal, who would oppose nationalisation.)
It isn't necessarily a good reflection on the EU, btw, if the larger countries can get away with things that smaller countries probably can't, but France flouted the eurozone's borrowing rules for years and nobody said a thing, cos it's France. Some specific ideas have been floated here such as that the EU would block any UK govt attempt to privatise the railway network that did not follow very specific, restrictive EU rules regarding tenders. It wouldn't. The UK could invoke national interest in taking control of infrastructure and just do it.