What is the current state of formalized mathematics?
On the one hand, there has been a lot of success recently in creating fully formalized and computer-verified proofs of nontrivial theorems, including Hales' theorem, the prime number theorem, the Jordan curve theorem, and Gödel's incompleteness theorems. The sense I get from experts in the field is that the main challenge is time, not theory. It takes a long time to formalize human-readable proofs into computer-verifiable proofs with the present systems, but there should be no theoretical obstacles to formalizing any theorem one wishes to study.
On the other hand, there are other reasons that nothing explicitly like Principia Mathematica has been developed. The first of these is that Principia is virtually unreadable. As a means of conveying mathematical information from one person to another, fully formal or even mostly formal proofs (the kind that a proof assistant can verify) are not as efficient as ordinary natural-language proofs. This means that few mathematicians have a desire to work with any "new" system of this kind. We already realize that virtually all mathematical theorems can be formalized in ZFC set theory, but instead we write proofs in a way that tries to convey the mathematical insight more than the technical details of a formal system, unless the technical details are somehow important.
There has been a lot of recent work on a different foundational system called "homotopy type theory", which could be used instead of ZFC to formalize theorems. It remains to be seen, however, whether this new system ends up being widely adopted. There other foundational systems, such as second-order arithmetic, which could also be used to fully formalize large parts of mathematics. I believe that a significant number of mathematicians don't really worry much about the foundational system they use, because the objects they deal with are sufficiently concrete that the foundations make little difference.
The other goal of Principia was to support the logicist program that all of mathematics can be reduced to logic. The idea that mathematics can be formalized and presented in full detail is no longer in question, as it might have been at the time. But the idea that the axioms of a foundational theory would all be fully logical is far from clear - in fact, it is generally considered false, because axioms such as the axiom of infinity or the axiom of replacement do not seem purely "logical" to many mathematicians.