Did Alan Turing's student Robin Gandy assert that Charles Babbage had no notion of a universal computing machine?
No, the opposite. This quote of Gandy's is not referring to Babbage, but to some intervening proposals for universal-style computing between Babbage and Turing. Gandy says those proposals did not have Babbage's recognition of the importance of branching and iteration to universal computation.



In "The Confluence of Ideas in 1936" by Gandy, as printed in the book "The Universal Turing Machine - A Half Century Survey", Section 2 is "Babbage and His Followers".

Here Gandy emphasizes that Babbage did understand and respect "conditional iteration" and "conditional transfer", e.g. end of p53 and top of p54


  Although Babbage mentions conditional transfer (67-68), he, with a natural respect for well-structured programming, uses only conditional iteration[....] He states conditional transfer explicitly (240), allowing that a 'go to' instruction may have to be executed by ringing a bell to summon the attendant; he gives an example of its use (241).


(Here Gandy refers to the article by Menabrea 1842 on Babbage's engine, but seems to attribute the ideas to Babbage himself.)

Gandy then quotes Babbage


  That the whole of the development and operations of analysis are now capable of being executed by machinery.


and writes


  Babbage, in his work on general algebra and functional equations, had shown
  his ability to think in abstract terms. If, then, one had led him to speculate (not
  difficult!) on what could be done with an abstract machine, free from limitations on
  its storage, he would surely have assented to a version (based on Sections 2.1.(1)-(5)) of Church's thesis.


Then Gandy goes on to Section 2.3, "Subsequent developments." He writes


  Other authors, concerned
  with more practical machines, referred to Babbage's work. Examples from Ran-
  dell 1982 are: M. d'Ocagne [1922], L. Couffignal [1933], V. Bush 1936, H.H. Aiken
  11964] (which is an unpublished memorandum of 1937). But the emphasis is on
  programming a fixed iterable sequence of arithmetical operations. The fundamental
  importance of conditional iteration and conditional transfer for a general theory of calculating machines is not recognized, though the principles may be used in very particular contexts [....]


Finally, Gandy writes:


  Conclusions. Babbage asserted what was, in effect a version of Church's thesis. His work was never entirely forgotten, but its theoretical importance - its importance, so to speak, as software - was little recognized[....]