Short answer: not much of a relation. Perhaps the stars could have aligned differently if the Vienna Circle lasted longer, but with the rise of Nazism many members had to flee Austria and the meetings effectively stopped in 1936. The majority of Vienna's associates were German speaking, and their focus was empirical science. There were not many mathematicians associated with Vienna, notably von Neumann, Menger and Wald, Gödel was close to Carnap in the late 1920s, but that is about it. The Bourbaki group was only coalescing in mid 1930s, and it mostly involved French mathematicians, so there was no personal connection. The focus of Bourbaki, at least originally, was reforming mathematics education, which also did not match the empirical and philosophical bent of Vienna.
Finally, it is only in the rear view mirror of today that everything formal and rigorous looks like a single lump. At the time, what moved Vienna and Bourbaki were very different drives, they descended from rival outlooks. The Vienna circle inherited Frege's approach, through Russell and (early) Wittgenstein, see e.g. Friedman, Logical Truth and Analyticity in Carnap's "Logical Syntax of Language". It favored genetic definitions (like Dedekind's of real numbers), universalist language (no meta), and "reducing mathematics to logic" (this is an oversimplification). Bourbaki, on the other hand, adopted (selectively) Hilbert's formalism with its pluralism of flexible axiomatics, and largely for pragmatic reason of not getting entangled with philosophy. On the deep divides between the those two visions of mathematics see e.g. Frege-Hilbert correspondence.
Gödel's results of early 1930s dealt a blow to both, but much more so to Fregeanism. Carnap did adjust to semantic paradigm in his later works, but the wind was taken out of the sails, so to speak. On the other hand, Bourbaki's pragmatic formalism light, with curbed ambitions for finitary justification, flourished. If the Vienna group survived into the times where the original foundational rivalries lost their significance perhaps there could have been fruitful interaction on the common ground of precision and rigor. But that was not to be.
Ironically, the "received view" of scientific theories associated with Vienna's Hempel, which was close in spirit to mathematical formalism, was displaced by the semantic view by 1970s, see Suppe, Understanding Scientific Theories: An Assessment of Developments, 1969-1998. One area that benefited from both influences was mathematical economics, see Martinelli-Luperi, The impacts of the early 20th century physics and mathematics crisis on contemporary economics discourse. The original Viena inspired works of von Neumann, Menger and his students on utility and general equilibrium was picked up by Debreu in 1950s, who attended Bourbaki seminars and was put off by "looseness" he found in contemporary economics texts.
