The equation
$$E = mc^2$$
says that energy is mass multiplied by speed of light squared.
This equation does not mention any particular unit of measurement, and holds irrespective of how we choose to measure energies, masses, distances, and time.
Units are only involved when we want to plug in actual numbers. For instance, we might know that in our experiment, $m = 1 \ \textrm{kg}$ and $c = 299792458 \ \textrm{m/s}$. Plugging that into our equation, we get:
$$ E = 1 \ \textrm{kg} \ (299792458 \ \textrm{m/s})^2$$
which simplifies to
$$ E = 89875517873681764 \ \frac{\textrm{kg} \ \textrm{m}^2}{\textrm{s}^2} $$
This gives us the energy in a rather esoteric looking unit of measurement. What if we wanted to express the energy in Joule instead? That is, we want to know which $x$ satisfies
$$ x \ \textrm{J} = E = 89875517873681764 \ \frac{\textrm{kg} \ \textrm{m}^2}{\textrm{s}^2}$$
This turns out a lot simpler than we might expect, because $J$ is defined as the energy imparted by a force of $1 \textrm{N}$ over a distance of $1 \textrm{m}$, and $1 \textrm{N}$ is defined as the force that accelerates an object of mass $1 \ \textrm{kg}$ by $1 \ \frac{\textrm{m}}{\textrm{s}}$ in $1 \ \textrm{s}$. 388
Putting all that together, we have:
$$\textrm{J} = \textrm{N} \ \textrm{m} = \textrm{kg} \frac{\textrm{m}}{\textrm{s}^2} \ \textrm{m} = \frac{\textrm{kg} \ \textrm{m}^2}{\textrm{s}^2}$$
Plugging this into our equation we get
$$ x \ \frac{\textrm{kg} \ \textrm{m}^2}{\textrm{s}^2} = E = 89875517873681764 \ \frac{\textrm{kg} \ \textrm{m}^2}{\textrm{s}^2}$$
which simplifies to
$$ x = 89875517873681764 $$
That is, it doesn't matter how long a meter is, or how long a second takes, or how much a kilogram of matter is, because a Joule is defined in terms of that same meter, that same second, and that same kilogram.
Put differently, the reason we don't need to do any unit conversion here is that our choice of units was not arbitrary, because we chose to measure energy in a unit that is derived from our chosen units for mass, distance and time.
This is, of course, intentional. The goal of the international system of units is to simplify calculations about the natural world, and eliminating unnecessary unit conversions by choosing coherent units of measurement definitely helps with that.
The flip side of striving for coherent systems of measurement is that systems of measurement must evolve alongside our understanding of the natural world. For instance, distance and time were thought to be unrelated until the speed of light was discovered to be constant. In a fully coherent system of measurement, we'd set this constant to 1 by measuring distances in light seconds rather than meters, which would simplify <PERSON>'s famous formula to
$$ E = m $$
However, since light seconds make for an unwieldy large unit of measurement for our everyday lives and most phenomena of interest to us, the SI system chose to keep the meter, but redefined it as:
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
thereby ensuring that the conversion factor is at least an integer :-)
In summary, while natural laws hold irrespective of units of measurement, calculations involving actual physical quantities can be simplified by using coherent systems of measurement, such as the international system of units. 388