If the speed of ball is large most of the time, then speed of air flow across the ball (as seen from reference frame attached to the ball) is large most of the time, which means that heat transfer from ball to air is predominantly due to forced convection rather than free convection. With this assumption, you may use forced convection relations for heat transfer, which depends on air-speed across the ball, surface temperature of the ball, and thermodynamic properties of air. However these relations are empirical and are usually for constant air-speed, while in your case air-speed is changing with time. You may assume some average value of heat transfer coefficient to get an estimate. If thermal conductivity of the ball is high enough that entire ball is nearly at uniform temperature at all times, then you won't have to solve for temperature field inside the ball (to find its surface temperature, which is what is required ultimately). Regarding this last point read up lumped-system analysis and Biot number.
The problem is quite complex to solve quantitatively and requires a differential calculus of multi-variable functions, but I'll try to simplify it.
Imagine that the object consist of many thin slices across the temperature gradient. Every second slice is a heat container with heat capacity $c \left[ \frac{J}{K}\right]$ and the remaining are heat conductors of a heat conductivity $h \left[ \frac{W}{K}\right]$ (just to separate two effects -- conductivity and capacity)
A the first instant all layers are at an ambient temperature. When you touch left side of the conductor, heat starts flowing, but the object is not at the equilibrium.
Heat flows into segment $A$ at the rate:
$$P_{A\,in}=h(T_{left}-T_A)$$
And the temperature of $A$ starts increasing.
$$\frac{dT_A}{dt} = \frac{P_{A\,in}-P_{A\,out}}{c}$$
When $T_A$ is just slightly over the ambient temperature it starts giving heat out to segment $B$
$$P_{A\,out}=P_{B\,in}=h(T_A-T_B)$$
And the temperature of $B$ starts rising which affects segment $C$, and so on.
This process continues until temperature gradient is established and heat absorbed and dissipated are equal.
The speed ot this process depends on density of the material, heat capacity, thickness, surface area, conductivity and much more. Although the process starts immediately, it takes some time until it becomes observable. In this case radiation does not contribute much to energy transfer since most conductors are opaque. Heat is mainly transferred through the collisions of the molecules.
I can show you derivation of temperature function for a single segment $A$ to illustrate the whole process.
$$P_{in}=h(T_{left}-T)$$
$$P_{out}=h(T-T_{amb})$$
$$\frac{dT}{dt} = \frac{P_{in}-P_{out}}{c} = \frac{h}{c}(T_{left}-2T+T_{amb})$$
$$\frac{dT}{T_{left}-2T+T_{amb}} = \frac{h}{c}dt$$
Solving this differential equation we obtain:
$$\frac{\ln(-2T+T_{left}+T_{amb})}{-2} = \frac{h}{c}t + C$$
Which, after some transformations, gives:
$$T = A \exp\left(\frac{-2h}{c}t\right)+\frac{1}{2}(T_{left}+T_{amb})$$
Where $A$ depends on initial conditions. In this case it is:
$$A=\frac{1}{2}(T_{amb}-T_{left})$$
You may notice, that for $t=0$, $T=T_{amb}$; and, after very long time, when equilibrium is established, $T$ is just between $T_{left}$ and $T_{amb}$.
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TL;DR: Heat can't travel instantaneously because relativity disallows it. Some microscopic models of heat transfer predict that heat travels in a specific formulation of the speed of the sound. So its kinda the "heat speed of sound" in the specific material
Heat can't travel instantaneously because it is limited through relativity by the speed of light. Yes, the Solution to the transient fourier equation allows for any small time that the temperature at the finite end of an object to increase, even if its just a tiny number, but is still nonzero, so in a sense, the fourier equation allows heat with unlimited speed, but that's not possible.
Its hardly a problem practically, though, because in practice the materials we utilize to measure heat transfer are small enough so that you can ignore relativistic effects. Only in, say , a bar that connects the earth and the sun, you would have to consider the speed of light limit.
But, there are some models that modify the fourier equation to be compatible with relativity. You can see them here: https://en.wikipedia.org/wiki/Relativistic_heat_conduction (of course, only to give you a very rough idea)