If I remember correctly, the notion of a model is already present in Hilbert-Bernays where finite structures are used for proving absolute consistency of some theories, and is probably older. Again, if I remember correctly, Frege, Russel, and Hilbert did have formal systems and the notion of a formal proof. Skolem's construction of a term model (which is now famous as Skolem's Paradox because the set of real numbers of the model turns out to be countable) is in his 1922 paper, where the Godel's completeness theorem is from 1929. In other words, it seems that Skolem did already have all the tools necessary for proving completeness in 1922. It seems that Hilbert had even stated the question of completeness for first-order theories before this date and Godel has learned about this problem in Carnap's logic course in 1928.
Hilbert's 10-th problem from his famous 23 problems asks for an algorithm to decide existence of solutions for Diophantine equations. I think there were attempts after this for understanding what is an algorithm. There were many definitions which came out before Turing's definition which were equivalent to his definition, although they were not philosophically satisfactory, at least Godel did not accept any of them as capturing the intuitive notion of computability before Turing's definition.
Godel's collected works can shed more light on these issues.
Solomon Feferman, "Gödel on finitism, constructivity and Hilbert’s program" http://math.stanford.edu/~feferman/papers/bernays.pdf
Hilbert and Ackermann posed the fundamental problem of the completeness of the first-order predicate calculus in their logic text of 1928; Gödel settled that question in the affirmative in his dissertation a year later. [page 2]
Hilbert introduced first order logic and raised the question of completeness much earlier, in his lectures of 1917-18. According to Awodey and Carus (2001), Gödel learned of this completeness problem in his logic course with Carnap in 1928 (the one logic course that he ever took!). [page 2, footnote]
Martin Davis, "What did Gödel believe and when did he believe it", BSL, 2005
Godel has emphasized the important role that his philosophical views had played in his discoveries. Thus, in a letter to Hao Wang of December 7, 1967, explaining why Skolem and others had not obtained the completeness theorem for predicate calculus, ... [page 1]