There is a subtle point here that should be noted: We have to know which halting problem we are talking about. If we are referring to the halting problem for Turing machines, that would mean that we can decide the consistency of today's axiomatic systems only. That is, mathematics may radically evolve if an algorithm that solves the halting problem for Turing machines was invented.
The reason why can be found in the definition of an axiomatic system. What we want is a set of axioms such that an algorithm decide whether a sentence is an axiom or not. Therefore new algorithmic processes may allow us to create new, more expressive axiomatic systems that cannot be coded in Turing machines.
And maybe this new class of algorithms will have similar properties to Turing machines and won't be able to solve its own halting problem. Then these new mathematics will likewise have uncertainties and require ingenuity in the discovery of theorems.