In summary, without a finite invariant speed, there would likely be no relativistic effects. This is because the principle of relativity, which is necessary for the existence of relativistic effects, would not apply. Without the principle of relativity, it is impossible to determine how the universe would look like, as it is based on hypothetical laws. Additionally, the only self-consistent theories that can be derived from the principle of relativity involve a finite invariant speed, making it a crucial factor in understanding relativistic effects. However, without the principle of relativity, it is impossible to determine the behavior of the universe, as demonstrated by the Mansouri-Sexl test theories. f
I assume that you mean "invariant speed" when you write "ultimate speed", which is not a term with which I am familiar.
It is possible to derive relativity from just the principle of relativity. You get two self-consistent theories, one with a finite invariant speed (Einstein's theory) and one with an infinite invariant speed (Newton's theory). In that sense, requiring a finite invariant speed selects the theory with relativistic effects over the theory without.
But note that I've added a lot to your question by assuming the principle of relativity. There is no particular reason to suppose that the principle of relativity applies, except that it works for our universe. Your question, taken generally, is just "how would physics be different if physics were different?". Such a general question cannot be answered, as mfb points out.
Would there have been any relativistic effects if there was no ultimate speed?
What do you mean by relativistic effects? Departures from Newtonian physics? Those are changes in the way we explain these effects. They are not changes in the way Nature behaves, just changes in the way we expect it to behave. People have been observing the way magnets attract iron for centuries, if not millennia. The fact that Einstein explained this as an effect of relative motion is a monumental accomplishment.
I don't know what the universe would be like if we didn't have things like magnetism. As far as I know, such a thing would not be possible.
Would there have been any relativistic effects if there was no ultimate speed?
Probably not. But it's difficult to be sure we share an understanding of what the word "relativistic effects" means. The reason I say probably not is based on the following mathematical argument (and a few vague memories). If we take the relativistic transform equation, the Lorentz transform
and formally take the limit as c -> infinity, we find that it reduces to the familiar non-relativsitc transform
$$x' = x - v\,t \quad t' = t$$
More generally, it's known that the Lorentz transform and the Galilean transform (which can be regarded as the formal limit of the former as c -> infinity as I just argued) are the only solutions compatible with the principle of relativity, which I'd describe as saying that there is no "special" preferred frame of reference. The arguments which show this are based on group theory, which informally I'd describe as resulting from the assumption that there is a description of physics in any frame of reference one chooses, and a 1:1 mapping between frames of reference.
Possibly if you were willing to violate the principle of relativity you could find alternatives, such as the Mansouri-Sexl test theories. I'd describe these theories as still having "ultimate velocities", but the ultimate velocities would be frame-dependent in such theories.