Your statement
working with and subtracting infinities ... which in general have no mathematical meaning
is not really correct, and seems to have a common misunderstanding in it. The technical difficulties from QFT do not come from infinities. In fact, ideas basically equivalent to renormalization and regularization have been used since the beginning of math--see, e.g., many papers by Cauchy, Euler, Riemann, etc. In fact, G.H. Hardy has a book published on the topic of divergent series:
http://www.amazon.com/Divergent-AMS-Chelsea-Publishing-Hardy/dp/0821826492
There is even a whole branch of math called "integration theory" (of which things like Lebesgue integration is a subset) that generalizes these types of issues. So having infinities show up is not an issue at all, in a sense, they show up out of convenience.
So the idea that infinities have anything to do with making QFT axiomatic is not correct.
The real issue, from a more formal point of view, is that you "want" to construct QFTs via some kind of path integral. But the path integral, formally (i.e., to mathematicians) is an integral (in the general sense that appears in topics like "integration theory") over a pretty pathological looking infinite dimensional LCSC function space.
Trying to define a reasonable measure on an infinite dimensional function space is problematic (and the general properties of these spaces doesn't seem to be particularly well understood). You run into problems like having all reasonable sets being "too small" to have a measure, worrying about measures of pathological sets, and worrying about what properties your measure should have, worrying if the "$\mathcal{D}\phi$" term is even a measure at all, etc...
At best, trying to fix this problem, you'd run into an issue like you have in the Lebesgue integral's definition, where it defines the integral and you construct some mathematically interesting properties, but most of its utility is in letting you abuse the Riemann integral in the way you wanted to. Actually calculating integrals from the definition of the Lebesgue integral is not generally easy. This isn't really enough to attract the attention of too many physicists, since we already have a definition that works, and knowing all of its formal properties would be nice, and would certainly tell us some surprising things, but it's not clear that it would be all that useful generally.
From an algebraic point of view, I believe you run into trouble with trying to define divergent products of operators that depend on renormalization scheme, so you need to have some family of $C^*$-algebras that respects renormalization group flow in the right way, but it doesn't seem like people have tried to do this in a reasonable way.
From a physics point of view, we don't care about any of this, because we can talk about renormalization, and demand that our answers have "physically reasonable" properties. You can do this mathematically, too, but the mathematicians are not interested in getting a reasonable answer; what they want is a set of "reasonable axioms" that the reasonable answers follow from, so they're doomed to run into technical difficulties like I mentioned above.
Formally, though, one can define non-interacting QFTs, and quantum mechanical path integrals. It's probably the case that formally defining a QFT is within the reach of what we could do if we really wanted, but it's just not a compelling topic to the people who understand how renormalization fixes the solutions to physically reasonable ones (physicists), and the formal aspects aren't well-understood enough that it's something one could get the formalism for "for free."
So my impression is that neither physicists or mathematicians generally care enough to work together to solve this problem, and it won't be solved until it can be done "for free" as a consequence of understanding other stuff.
Edit:
I should also add briefly that CFTs and SCFTs are mathematically much more carefully defined, and so a reasonable alternative to the classic ideas I mentioned above might be to start with a SCFT, and define a general field theory as some kind of "small" modification of it, done in such a way to keep just the right things well-defined.
The three classes of QFTs you are referring to are distinguished by different symmetry assumptions (Poincare invariance, conformal invariance, and volume-preserving diffeomorphism invariance) and different background spacetimes (Minkowski, Riemann curve (or families of them), and arbitrary manifolds). Moreover, Wightman axioms only characterize the vacuum sector of a Poincare-invariant QFT.
Each set of assumptions leads to very different natural questions and constructions, hence different mathematical approaches. This explains why there is this diversity of approaches. In the light of this diversity, a uniform theory would be conceptually very shallow - too general to be restrictive and hence useful - and would break immediately into chapters distinguished by specific assumptions.
Best Answer
CFT consists for most mathematicians - who are interested in this topic - currently of the study of vertex operator algebras, see this question on math overflow:
You can find a little bit more about the topic and the work of several mathematicians here:
As you can see from the answers on mathoverflow, vertex algebras were not invented for the study of CFT, and that they form an axiomatic abstraction of operator algebra products was noted only later.
A personal and very subjective note: One should not underestimate the amount of theoretical physics that is necessary to understand what a QFT is to physicists. Most mathematicians that encounter physics for the first time since highschool through some QFT framework seem to be quite taken aback by the high intrance fee they'd have to pay to understand this. This is my own, personal explanation for the observtion that most mathematicians study the formal machinery only, in order to use it to prove some new mathematical theorems, but only very rarely in order to better understand what physicists do. Although you'll find quite a lot of work by quite a lot of rather famous mathematicians if you follow the links above, AFAIK there is none doing the work you describe.