A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bologna, 1921 & 1923) --- I have not found an English translation, but an English summary by Tonelli is also in an AMS bulletin. It is noteworthy that Tonelli himself, in this summary but also in other writings, does not mention Hilbert at all. Apparently he considered his contributions to be quite independent.
A more recent discussion of Tonelli's work in the context of Hilbert's 20th (and 19th) problem is given by Francis Clarke (2008):
The decade preceding the formulation of Hilbert’s problems had been
marked by a controversy over the Dirichlet principle, which affirms
the equivalence between functions $u$ minimizing the Dirichlet
functional $J$ and solutions $u$ of Laplace’s equation. As Weierstrass
and Hilbert pointed out in response to (notably) Riemann’s assertions,
the existence of a minimum here (and the very class in which to seek
one) is problematic. Hilbert went on to give the first rigorous
treatment of the issue in 1904, in a context which succeeded in
limiting the class of functions $u$ involved to Lipschitz ones. But it
became clear that a more general type of function space was needed,
and finally Sobolev spaces provided a suitable context in which to
assert the existence of a solution to the basic problem.
The direct method introduced by Tonelli exploits the weak sequential
compactness of a minimizing sequence and the weak lower semicontinuity
of the convex functional $J$, to deduce the existence of a solution
$u$ in the Sobolev space $W^{1,1}$, as well as the uniqueness (since
$J$ is strictly convex), answering Hilbert’s 20th problem. The
remaining problem (Hilbert's 19th) was the regularity of the solution
$u$, especially since functions in the Sobolev space $W^{1,1}$ are not
even continuous necessarily.
