One possible such theory is described by Solomon Feferman in chapter 13 (“Weyl vindicated: Das Kontinuuum seventy years later”) of his book In the Light of Logic (Oxford University Press 1998); specifically, I am referring to the formal system denoted $W$ and described in §8 (“A Theory of Flexible Finite Types for Weyl's Program”) of the aforementioned chapter.
This system $W$ is intended by Feferman to be a precise formalization of a proposed foundation for analysis set forth in Hermann Weyl's 1918 monograph Das Kontinuum, kritische Untersuchungen über die Grundlagen der Analysis.
It is analyzed metamathematically in two papers by Feferman & Jäger, “Systems of explicit mathematics with non-constructive $\mu$-operator” parts I & II (1993 & 1996), in which it is proved (IIUC, because the notation is very confusing) that $W$ is a conservative extension of $\mathrm{PA}$.
In §9 of the same chapter, as well as in the next chapter (“Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics”), Feferman argues that “scientifically applicable” mathematics can be formalized in $W$, and discusses some possible issues, and the relation with other foundational frameworks and ideas.
(I think there is also a constructive version of $W$, which would make sense as Weyl was IIUC at least somewhat sympathetic to constructive mathematics, but Feferman seems to only allude to it, so I don't know if it has been described or studied in any detail. I'm also unsure about the relation between the system $W$ described in the aforementioned 1998 book and the one described in another of Feferman's papers, The significance of Hermann Weyl'sDas Kontinuum— which is also based on Weyl's monograph, but maybe not exactly identical to $W$ in all details, and which he mostly compares to $\mathrm{ACA}_0$. I'm afraid I can't say much more, because I only looked at this a long time ago and not too carefully even then, and now I'm very confused as to the exact relation between the various systems described by Feferman. I'm not even sure whether there is consensus on whether this is a faithful reformulation of Weyl's intended system.)