First is the speed of light is not the speed of light, it is the speed of causality. PBS Space Time has a great video about this. Light goes at this speed because that's as fast as events are allowed to propagate in our universe. Make the speed of light infinite and now the speed of causality is infinite. This means events which happen here can affect distant parts of the universe instantaneously. This has serious consequences.
The speed of light/causality is also a sort of universal conversion factor between mass and energy, and space and time. If it's infinite then mass, energy, space, and time are now four separate things, not two. This also has serious consequences.
In our universe we have 4 dimensional "spacetime". A Newtonian universe has 3 dimensional "space" and 1 dimensional "time". The speed of light is the conversion factor between space and time. If it's infinite there's no conversion. Space and time are now fundamentally different things. This has deep consequences for how our universe works, such as gravity. For example, gravity is not a force, but it's curved 4D spacetime. This solved many small discrepancies in planetary and stellar observations. That's easily handled by saying discrepancies simply wouldn't exist in your universe. Ok, but astronomy is really the least of the problems.
Mass and energy would be different things.
E = mc^2, the mass-energy equivalence, would not be. This has deep, deep consequences. The mass-energy equivalence doesn't say that mass can be converted into energy, it says mass is a property of energy. There is no such thing as "matter", it's just a special form of energy. So much of our physical world relies on this. In a Newtonian universe matter and energy are separate things. Here's more about the significance of
E = mc^2 from PBS Space Time.
Since matter and energy are now non-convertible, both energy and matter would always be conserved. Fusion, which converts part of the rest mass into energy, would not be possible. No fusion, no heat from stars. No heat from stars, no life. Stars just collapse under gravity into lumps of matter initially heated by gravitational collapse but eventually going cold. Neutron stars (and regular stars) require quantum effects which don't exist in a Newtonian universe.
In a relativistic universe, because of the mass-energy equivalence, energy has "mass". A box containing 1 gram of matter and one containing 9x1013 J of energy have the same mass-energy and the same gravitational pull. In a Newtonian universe this is not so. There is no mass-energy equivalence so energy does not exert nor feel gravity. No gravitational lensing. No black holes. Energy can only be absorbed and reflected.
The speed of light shows up in many equations, even those they appear not to. Many of the physical equations we learn in school are actually non-relativistic versions only for use at low relative velocities. For example, the kinetic energy equation:
1/2 m * v^2 is far more complicated in relativity and uses the speed of light. Making the speed of light infinite changes all this.
Then there's the question of time. If light, energy, and causality travel at infinite speeds, we have no "past" and no "future". Everything happens in the "present" all at once. If the speed of light/causality is infinite, we have no time. Again, here's PBS Space Time on the origin of time and why speed of light is necessary for time to exist.
There you have it. No space-time. No mass-energy equivalence. No fusion. No stars. No time. A Newtonian universe would be very, very, very different.
That's just the broad overview. Many, many everyday equations change, or must be derived in a radically different way. I also didn't get into the lack of quantum mechanics which eliminates everything from transistors to stars (again). There's more about that on Physics.SE. For your universe to work and be consistent you'd have to re-engineer a significant portion of reality so the macroscopic world still works as it does now.
Alternatively you leave your world at the 18th century level of understanding. Stars shine, there is time and causality, they don't know how or why, and you don't need to explain it. The linear relationship between velocity and kinetic energy allows spaceships with far simpler technology.
Or you just don't let anyone with a decent physics background play. :)