Such a timeline is necessarily highly subjective.

With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory.
The list below concentrates on developments that in some way affect the foundations of homotopy theory, as opposed to general advances in homotopy theory.
Given the length of the list, I probably omitted many important developments,
feel free to point them out in the comments!
I also excluded from consideration the last decade or so, restricting to older developments.

**Poincaré** defined _homology_ (via Betti numbers) and the _fundamental group_ in a series of papers starting from 1895.
The initial approach was nonrigorous, but in response to the resulting criticism,
Poincaré reformulated his work in terms of simplicial complexes.

**Fréchet** defined _metric spaces_ in 1906 and **Hausdorff** defined _topological spaces_ in 1914.
This enabled the study the topological properties of spaces without first triangulating them.

Around 1925, Emmy **Noether** proposed to upgrade Betti numbers to _homology groups_.
In connection with this, sometime in 1930s, the terminology shifted from “combinatorial topology” to “algebraic topology”.

Around 1931, **Veblen** and J. H. C. **Whitehead** introduced the modern definition of a _smooth manifold_.

**Eilenberg** defined _singular homology_ in 1943,
which resulted in a systematic study of homology and cohomology (defined by **Kolmogoroff** and **Alexander** in 1936)
of arbitrary topological spaces.

**Eilenberg** and **MacLane** introduced _categories_, _functors_, and _natural transformations_ in 1945.
Ever since then, category theory played an increasingly important role in homotopy theory,
to the point where we are now often unable to cleanly separate them.

**Eilenberg** and **Zilber** developed the theory of _simplicial sets_ (known at the time as “complete semi-simplicial complexes**) in 1949.

J. H. C. **Whitehead** proved what is now known as the _Whitehead theorem_ in 1948.

**Eilenberg** and **Steenrod** published their _Foundations of Algebraic Topology_ in 1952,
formulating what is now known as the _Eilenberg–Steenrod axioms_.

**Kan** (advised by **Eilenberg**) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory)
starting from around 1955.
He introduced combinatorial homotopy groups, the **Dold**–Kan correspondence,
adjoint functors, limits and colimits, Kan extensions, etc.

**Lima** defined _spectra_ in 1958.

**Quillen** published his _Homotopical Algebra_ in 1967, introducing _model categories_
and using them in his _Rational homotopy theory_ around 1968.
Around 1972, he introduced _higher algebraic K-theory_.

**Segal** introduced <a href="https://doi.org/10.1016/0040-9383(74)90022-6">_Γ-spaces_</a> around 1972.

**Brown** studied the homotopy theory of <a href="https://doi.org/10.2307/1996573">_sheaves of spaces and spectra_</a> in 1972.

**Boardman** and **Vogt** introduced <a href="https://doi.org/10.1007/BFb0068547">_quasicategories_</a> in 1973.

**Dwyer** and **Kan** introduced and developed the theory of <a href="https://doi.org/10.1016/0022-4049(80)90049-3">_simplicial localizations_</a> starting from around 1979.

Around 1979, **Bousfield** introduced what is now known as <a href="https://doi.org/10.1016/0040-9383(79)90018-1">_Bousfield localizations_</a>.

In 1980s, **Joyal** established what is now known as the <a href="https://ncatlab.org/nlab/show/model+structure+on+simplicial+sets#joyal_model_structure_on_simplicial_sets">_Joyal model structure_</a> on simplicial sets.

In mid-1980s, **Segal** (following Witten) introduced what is now known as <a href="https://doi.org/10.1007/978-94-015-7809-7_9">_functorial field theory_</a>,
later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.

In 1985, **Jardine** gave an account of <a href="http://math.uchicago.edu/~amathew/simplicialpresheaves.pdf">_simplicial presheaves_</a>.

Around 1986, **Lewis**, **May**, **Steinberger**, **McClure** introduced genuine <a href="https://doi.org/10.1007/BFb0075778">equivariant spectra</a>.

In 1995, **Baez** and **Dolan** formulated the <a href="https://arxiv.org/abs/q-alg/9503002">_cobordism and tangle hypotheses_</a>, which perhaps qualifies as the first noticeable conjecture
about (∞,n)-categories for arbitrary n.

In 1997, **Elmendorf**, **Kriz**, **Mandell**, **May** published the first ever account of a <a href="https://web.archive.org/web/20161019233205id_/http://www.math.uchicago.edu:80/~may/BOOKS/EKMM.pdf">_symmetric monoidal category of spectra_</a>.

In 1998, **Hovey**, **Shipley**, **Smith** published an account of <a href="https://arxiv.org/abs/math/9801077">_symmetric spectra_</a>.

In 1998, **Rezk** introduced <a href="https://arxiv.org/abs/math/9811037">_complete Segal spaces_</a>.

Around the late 1990s, **Smith** introduced _combinatorial model categories_
and proved what is now known as the <a href="https://ncatlab.org/nlab/show/combinatorial+model+category#SmithTheorem">_Smith recognition theorem_</a>
and established the existence of left Bousfield localizations of left proper combinatorial model categories.

<a href="https://arxiv.org/abs/math/0209342">_Monoidal model categories_</a> were systematically studied by **Schwede** and **Shipley** starting from the late 1990s.

In the mid-2000s, **Lurie**'s <a href="https://arxiv.org/abs/math/0608040">_Higher Topos Theory_</a> came out, first as an online draft, which was later published.