How could we define speed without time?

This is a bit of a brain twister but it's a very serious question. Please be precise in your answer.

I do not believe time exists. I believe it is simply an illusion of our perception, the perception of seeing things around us change. The faster things change the more time seems to go by. In some sort, change is the result of the dissipation of energy. Thus what I believe is that what we are really observing is the energy being dissipated and created all around us and we call it time.

Since time doesn't exist (for me). That would mean that the equation of [ speed = distance / time ] is impossible. But nevertheless the concept of speed is fundamental for physics. What could we replace this equation with to no longer have speed dependent of time but of energy?

Answer

We know that to position objects in spacetime requires four coordinates e.g. $(t, x, y, z)$. So time certainly exists. The point you're addressing is about the flow of time. Incidentally this point is discussed in some detail in the question Is there a proof of existence of time?.

Any object traces out a worldline that is a curve in spacetime, and we can parameterise this curve by using an affine parameter $\tau$ that varies along the curve, then write the coordinates as a function of this parameter, $t(\tau)$, $x(\tau)$, $y(\tau)$ and $z(\tau)$. In fact this is exactly what is done in General Relativity. In GR the affine parameter is normally the proper time, but any parameter can be used and need not have any physical significance. For example photon world lines can be parameterised in this way even that the proper time is everywhere constant for a photon.

Having done this we can now calculate values for $dx/dt$, $dy/dt$, etc and even things like $dx/dy$ if we wish. Then we can define a coordinate velocity as:

$$ v^2 = \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2$$

So the point is that we can calculate a velocity without worrying about whether there is a flow of time in the sense humans normally use the term. There is no need to replace velocity with any other quantity.