That the two nations shared monarch - James VI of Scotland and I of England - who conducted foreign policy on behalf of both was a blow to the balance of power in Western Europe.
Traditionally, Scotland had maintained close relationships with France ("the Auld Alliance") as a counterweight to its powerful Southern neighbour. James VI/I wasn't out to alienate France - he had strong links there, his godfather was King Charles IX, his grandmother was the French-born Mary of Guise and his mother, Mary, Queen of Scots, had lived in France and was briefly Queen Consort to King Francis II - but he was acutely aware of the English parliament and nobility's enmity, as well as the vocal anti-Catholic factions in both Scotland and France.
James VI/I ended the war with Spain and estranging England from its Protestant allies in the Dutch United Provinces - a victory for pragmatism over zeal, but one that infuriated the French who had also been at war with Spain and were spoiling for a rematch..
Dudley Carleton, parliamentarian and future ambassadors to Venice and the United Provinces, recalled opposition from both the French and Spanish ambassadors to London - singling out French ambassador Christophe de Harlay, Comte de Beaumont, especially as critical, (rightly) viewing it as the end of the Auld Alliance and complaining that the court was awash with Spaniards.
I'll leave the Act of Union and the French-backing of the Jacobite cause to someone else!
Sources: Scots and Britons: Scottish Political Thought and the Union of 1603 by Roger A Mason and Drama and the Succession to the Crown, 1561–1633 by Professor Lisa Hopkins.