I've been avoiding answering this question, as I think axiomatization of time is a rather fruitless activity. But, as no one else has mentioned this, I feel, unfortunately, compelled to post this answer.

The axiomatic approach to topological quantum field theory [Blanchet and Turaev] defines a topological quantum field theory,

Definition (TQFT): A $(n + 1)$-dimensional TQFT $(V,\tau)$ over a scalar ﬁeld $k$ assigns to every closed oriented $n$-dimensional manifold $X$ a finite dimensional vector space $V(X)$ over $k$ and assigns to every cobordisim $(M,X,Y)$ a $k$-linear map $\tau(M) = \tau(M,X,Y ):V(X) \rightarrow V(Y )$.

In addition, the axiomatic approach to topological quantum field theory contains the normalization axiom,

Axiom (Normalization Axiom): For any n-dimensional manifold $X$, the linear map $\tau([0, 1] \times X) : V(X) \rightarrow V(X)$ is identity.

This normalization axiom is an axiomatization of time as it occurs in any diffeomorphism invariant theory.

In more detail, any theory that is diffeomorphism invariant is, in particular, invariant with respect to diffeomorphisms in the time direction $t'(t)$. The generator of time evolution is the Hamiltonian $H$. Thus, any state in the Hilbert space is invariant under the action of the Hamiltonian $H$. This is the exact content of the normalization axiom.