Since you don't mention any type of thermal contact resistance, I'm going to ignore that aspect and assume that the contact is thermally perfect.
As @Gert notes, the bodies must be initially separated to have different temperatures at their surfaces, as a single point or plane can have only one well-defined temperature. At the instant of contact, we find that conduction, convection and radiation across the quickly narrowing gap rapidly force the surfaces to assume the same interface temperature.
Immediately after contact and for a while, the interface temperature $T_\text{int}$ is
$$T_\text{int}=\frac{T_\text{a}(x,0)\sqrt{k_\text{a}\rho_\text{a}c_\text{a}}+T_\text{b}(x,0)\sqrt{k_\text{b}\rho_\text{b}c_\text{b}}}{\sqrt{k_\text{a}\rho_\text{a}c_\text{a}}+\sqrt{k_\text{b}\rho_\text{b}c_\text{b}}},$$
where $k_i$ is the thermal conductivity of material $i$, $\rho_i$ is the density, and $c_i$ is the constant-pressure specific heat capacity. Note that for identical materials, the interface temperature lies halfway between the two initial temperatures, which we'd expect by symmetry.
Interestingly, this interface temperature is a fixed value that depends only the material properties (here assumed to be temperature-independent) and the initial temperatures! At the interface, we predict a steadily decreasing heat flux—but constant temperature—as the initially sharp temperature gradient evens out:
Proof (adapted from Incropera & DeWitt's Fundamentals of Heat and Mass Transfer):
The temperature within each body satisfies the conduction heat equation
$$\alpha_i\frac{\partial ^2 T_i(x,t)}{\partial x^2}=\frac{\partial T_i(x,t)}{\partial t},$$
where $\alpha=\frac{k}{\rho c}$ is the thermal diffusivity. At the interface (taken as $x=0$), for time $t>0$, we have the boundary conditions
$$T_\text{a}(0,t)=T_\text{b}(0,t)=T_\text{int};$$ $$k_\text{a}\frac{dT_\text{a}(0,t)}{dx}=k_\text{b}\frac{dT_\text{b}(0,t)}{dx}.$$ The first condition reflects continuity of temperature at a single point; the second reflects an energy balance using Fourier's law of conduction: the fluxes $q^{\prime\prime}=-k_i\frac{dT_i}{dx}$ at the interface must be equal.
For a while after contact, the interface doesn't yet "know" about the other ends of the bodies (and vice versa). In other words, at times less than approximately the rule-of-thumb diffusion time, $\frac{L^2}{\alpha}$ (where $L$ is the body length), the ends are only minimally affected by the joining process and in turn can have little thermal effect on the interface.
(As a specific example, if you were to heat the very end of a 1 cm and 2 cm steel sample, you'd see the same spatiotemporal temperature distribution for the first few seconds for both samples, until the heat wave substantially diffused 1 cm and then started to respond to the length difference.)
Thus, for a while, we can idealize the bodies as semi-infinite, which eases the solution process substantially. In boundary-condition terms, we express this as $T_\text{a}(x\to\infty,t)\to T_\text{a}(x,0)$, $T_\text{b}(x\to-\infty,t)\to T_\text{b}(x,0)$.
We introduce the similarity variable $\eta_i=\frac{x}{2\sqrt{\alpha t}}$, whereupon the heat equation becomes
$$\frac{\partial ^2 T_i(\eta)}{\partial \eta^2}=-2\eta\frac{\partial T_i(\eta)}{\partial \eta}.$$
Integrating twice, we obtain
$$T_i(\eta_i)=C_1\int_0^{\eta_i}\exp(-\eta^{\prime2})\,d\eta^\prime+C_2$$
(where $\eta^\prime$ is a dummy variable of integration) and ultimately
$$T_i(x,t)=T_i(x,0)+[T_\text{int}-T_i(x,0)]\,\text{erfc}\frac{|x|}{2\sqrt{\alpha_i t}},$$
where $\text{erfc}$ denotes the complementary error function. For small $x$ or large $t$, $T_i(x,t)\approx T_\text{int}$; for large $x$ or small $t$, $T_i(x,t)\approx T_i(x,0)$, with a smooth transition between the two extremes.
The corresponding flux is $$q^{\prime\prime}=-k_\text{a}\frac{dT_\text{a}}{dx}=\frac{k_\text{a}(T_\text{int}-T_\text{a}(x,0))}{\sqrt{\pi\alpha t}}.$$
Setting the fluxes equal as described above, we obtain the short-time interface temperature $T_\text{int}$ given above.
Starting around the approximate characteristic time $\frac{L^2}{\alpha}$ of thermal diffusion down the bodies, and over many multiples of this characteristic time, the interface temperature transitions to the final uniform temperature of the two bodies
$$T_\text{int}=\frac{T_\text{a}(x,0)C_\text{a}+T_\text{b}(x,0)C_\text{b}}{C_\text{a}+C_\text{b}},$$
where $C_i$ is the heat capacity of body $i$. Note that for identical bodies, the interface temperature remains halfway between the two initial temperatures, which we'd again expect by symmetry.
This long-time interface temperature is obtained simply from a steady-state energy balance of both bodies. The interpretation here is that the diffusion process has come to the physical end of the hotter body that supplies heat flux and the end of the colder body that absorbs it. The temperature gradients that persisted within the bodies now vanish, there being nothing to sustain them. The final temperature, at steady state with zero heat flux, depends on the bodies' heat capacities (but not their thermal conductivities, as no fluxes remain).
