Gauss' posthumous publications? I'm looking for any information about the posthumous publication of Gauss' mathematical correspondence and notebooks.
When did these become widely available, and how did it affect progress in mathematics?
 A: The published version of Gauss's collected works, which contains 12 volumes (together with two other volumes that contain treatises by later mathematicians about different aspects of Gauss's work), can be accessed freely on the digital library of Gottingen university - here is a link: https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348. This version contains all of Gauss's publications, together with a lot of unpublished manuscripts, drafts of treatises and etc.
In addition, there are published correspondences of Gauss with different mathematicians/scientists with whom he corresponded regularly. These correspondences (for example: Gauss-Schumacher, Gauss-Bessel,...) fill many more volumes but their quality (in terms of density of mathematical ideas) is much lower than in Gauss's collected works mentioned before.
And finally, and this is something that I'm especially interested in, this is not the whole story - there are many fragments and notebooks of Gauss that have not been digitized yet - here is a link to a website that apparently gives a complete guide to Gauss's work (published as well as unpublished) - https://kalliope-verbund.info/de/findingaid_toc?fa.id=DE-611-BF-61709&fa.enum=1&lastparam=true. Note that some references that you will find in scientific/mathematical articles about Gauss cannot be located within the twelve volumes mentioned before. For example, the famous handnote in which Gauss began to study the topological object known as "braid" - which is the theme of another Mathoverflow question - cannot be found anywhere in his collected works, but according to the guide in the last link, it should be found in Handbuch 7.
Effect on the progress of mathematics
As a source of inspiration
Most of Gauss's unpublished discoveries were independently rediscovered by other mathematicians, albeit in a much later date, before the publication of his collected works. However, even in the case of already known results found in his Nachlass, it is not a specific discovery that affected the progress of mathematics, but rather the organic structure of his body of mathematical work, that served as a source of inspiration for later mathematicians. Once a more complete view of his work was available to the world, it was the interaction between his ideas in different branches of mathematics/science he worked in that fertilized the work of later researchers.
For example, there are deep relations between his work in differential geometry (especially his study of geodesics) and analytical mechanics (a branch of which he contributed his "principle of least constraint") as developed by later mathematicians. However, Gauss himself does not make any remark on such a connection, and Heinrich Herz gave much later an interpretation of Gauss's principle of least constraint using differential geometry. This is just an example to illustrate my point.
Examples of unpublished discoveries that were actually not known
Number theory

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*Two such examples were already mentioned in the answer above - the FFT algorithm, and the discoveries in Gauss's diary. Diary entry 146 was quite a remarkable result on the arithmetics of a certain elliptic curve that was unknown to the wider mathematical community.

*Gauss's geometric proof of biquadratic reciprocity law - this was a highly original proof that was very different from Eisenstein's analytic proof of this theorem. However, the development of number theory in the 19th century moved away from such geometric considerations, and as a result Gauss's proof remains largely unknown (even today). I don't think it had any influence on the progress of mathematics.

Analysis
From my experience at reading into Gauss, I gained the impression that it was especially the analytic aspects of his work (elliptic functions, theta functions, modular forms) that did have an impact on the progress of mathematics:

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*Gauss's posthumous treatise on the arithmetic geometric mean, that contained a lot of results that were unknown.

*When Richard Dedekind introduced the modular j-invariant, he was reffering  in particular to certain passages in Gauss's Nachlass that show he also intended to introduce such a function. Gauss was aware of its basic properties as well as its fundamental domain, but this was already discovered by later great mathematicians such as Jacobi and Riemann.

*Gauss's Nachlass contained the first ever drawing of an hyperbolic tessellation of the unit disk, which shows a uniform tilling of the unit disk by "equilateral" hyperbolic triangles with angles $\pi/4,\pi/4,\pi/4$. This sketch must surely have influenced Dedekind's and Klein's work. In particular, Gauss's results on the metrical relations in this network of curved triangles show he had anticipated the Poincare disk model and the Cayley-Klein metric associated with it.

Topology

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*Gauss's linking integral, an analytic method for calculating linking index of two closed space curves, was also widely unknown by the time of its publication. Maxwell independently rediscovered it about the same time Gauss's fragment was published, so I guess Gauss's fragment did make an effect.

*Gauss's ideas on planar "tract figures", which include a reference to an "algorithmic word problem", apparently had influence, as is evident in Max Dehn's paper "das Gaussische Problem der Trakte".

A: Q1: The mathematical diary that Gauss kept from 1796 to 1814 was rediscovered in 1897 and published in 1903, so almost fifty years after his death. His collected works were published sooner, in 1866.
Q2: According to The Poincaré Conjecture: In Search of the Shape of the Universe (page 124) the posthumous publication of Gauss's correspondence and scientific notebooks made it clear that Gauss had discovered non-Euclidean geometry first, and hastened the acceptance of Bolyai's and Lobachevsky's work.

As an aside: A notable discovery in Gauss' posthumous collected works was the basic algorithm of the fast Fourier transform, which he had already written down in 1805 -- even before Fourier's work from 1822. The FFT was not rediscovered until 1965. Other examples of independent rediscoveries include the Gauss-Seidel method and the quaternion multiplication rule.

A: I found a good source of information:
A Critical Survey and Inventory of the Edited Works of Carl Friedrich Gauss
https://link.springer.com/chapter/10.1007/978-3-319-73577-1_8
