The tone could not have been more different in Paris yesterday. Gone was the pomp and showbiz of Nicolas Sarkozy, in its place the almost Spartan inauguration of France's first Socialist leader for 17 years. And as the latest figures showed the French economy stalling and only German expansion keeping the eurozone out of recession, François Hollande took the opportunity to burnish his credentials as Europe's anti-austerity cheerleader, promising an "essential" economic stimulus alongside the admitted necessity of cutting public debt.

For Britain, the new man in the Elysée palace – for all his left-wing politics – will make little immediate difference. True, there may be a wrinkle over military co-operation, already knocked by Britain's decision to buy vertical take-off F35s (which rules French Rafale jets out of using our aircraft carriers) and facing further strain if the Socialists in Paris look to their defence budget when searching for cuts. In broad terms, however, the long-standing sympathies, and differences, between the two countries remain much as ever they were.

It is Mr Hollande's role in Europe that will define his presidency, as his almost immediate departure for Berlin yesterday attests. The meeting has been billed as a clash of fundamental opposites, the centre-right German Chancellor's unwavering fiscal rectitude pitted against Mr Hollande's left-leaning desire to prioritise growth. In reality, while it may suit both to present themselves that way to their domestic audiences, the divergence between them is as much one of emphasis as of substance.

Where Angela Merkel favours structural reform, for example, Mr Hollande prefers infrastructure investment financed by EU "project bonds". Both positions can be accommodated within the existing fiscal treaty without much difficulty. The German Chancellor might even be grateful for a new European partner able to provide vital political cover for a shift it would be unwise, and impractical, for her to lead herself.

For all his warm words, Mr Hollande is far from radical even with regard to his domestic concerns. He may have larded his election campaign with pledges to tax the rich, splash out on teachers and cut back the retirement age. But he has also explicitly committed himself to his predecessor's most imminent deficit-reduction targets, and only shifted the deadline for balancing the budget altogether by a single year. The only question now is how he can hope to afford it all.

The mood music in France and Germany may be of a different tempo, therefore, but the song is largely the same. Just as importantly, the personal chemistry of the partnership at the heart of the eurozone may now work better than it did. They may come from different sides of the political spectrum, but Ms Merkel will likely have more in common with Mr Hollande's sober, consensus-building and old-style European leanings than ever she had with Mr Sarkozy's tempestuous flamboyance and outspoken Atlanticism.

It is here, in fact, in the smoke and mirrors of political positioning, that the French election has the greatest significance for the British Government. Not only has David Cameron lost an ally on the centre-right, and Labour received a fillip for its anti-austerity message. The Socialist victory in France – taken together with recent wins for the German left in two regional elections – only adds to the growing sense that the tide is turning against the untrammelled fiscal discipline by which the Coalition defines itself. It matters little whether the explanation is a genuine ideological shift or merely a spasm of anti-incumbency. Mr Cameron cut himself off from Europe with his shock veto of the fiscal treaty late last year. Mr Hollande's inauguration leaves him more isolated than ever.