The following probably isn't what you have in mind, but it does raise a point that is very relevant to the physical content, as opposed to the mathematical content, of your question.
The fact is that many lattice theories are used everyday by physicists in calculating numerical approximations to the continuum theories we believe they approximate. All computer modelling outputs the results of discrete, difference equations and in many fields of modern physics there simply are no analytical solutions to replace them. Lattice QCD, numerical general relativity and numerical fluid dynamics are excellent examples as there is almost no other practical way of calculating what these theories actually foretell. And the experimental observation (as well as the theoretical reality for many physical theories) is that the predictions made by discretized theories are excellent models both of reality and of corresponding continuous models where these latter exist - often to well within experimental error. So here is the crucial point that applies to your question:
Whether the underlying theory should be a continuous model, or a discrete system, is a proposition that so far cannot be tested by experiment. Continuum and discretized models yield the same results as the discretized theory deals with finer and finer meshes
So I think in a very real sense, so far your question really doesn't belong to physics as it cannot be answered experimentally. And there doesn't seem to be any prospect on the horizon of its being so.
So the physics answer is quite different from the mathematical one, which amounts to whether the universe is a countable, or even finite, collection of fundamental "atoms" as opposed to an uncountable continuum. The uncountable / countable distinction is utterly real (and extremely important) in mathematics, but so far we haven't found a way that such a distinction might express itself as an experimental, measurable result.