In Differential Geometry (a branch of mathematics) there exists the notion of an umbilical point. Is there a noun corresponding to the adjective umbilical? Could I write something like "It follows by umbilicality that […]"?
Umbilical in this sense is an adjective, not a verb (umbilical is also used as a noun synonymous with umbilical cord in both the biological sense and several figurative senses analogous to it). In mathematics, there is the noun umbilic synonymous with umbilical point.
It is quite correct to create the noun umbilicality to mean the quality held by something which is umbilical, in any sense of the word. This has been done before in the mathematical sense, and in other senses at least as far back as the 17th Century (Sir Thomas Browne, Pseudodoxia Epidemica or Enquries into very many received tenets and commonly presumed truths, 1646).
Looking up umbilic in terms of geometry, finds it given as a synonym of umbilical point, and of umbilical as it relates to those. Since in this sense there's no tendency - as there is in other senses - to favour umbilical for the cord and umbilic for anything else related to the navel, then umbilicity would have just as much justification as umbilicality. Since it's also shorter, less clumsy sounding and - most importantly of all - already used by many in this context, it would be the one to go for.
There's nothing wrong with deriving umbilicality from umbilical. The same process generates mortality, sexuality, speciality, personality, for example.
I don't know much about either obstetrics or OP's specialised context, but I guess he wants it to mean principles relating to "umbilical" entities/relationships in mathematics. Nothing wrong with that either - as the above list shows, the precise meaning of the derived word isn't always exactly the state of being xxxxx-al.
But there's no particular grammatical/liguistic principle saying the noun "should" be umbilical as opposed to, say, umbilicity. Come to that, if we really did need a word meaning the state of being umbilical, umbilicalness would be just as valid as umbilicality.
Admittedly it's a bit old (1843), but this citation from OED is perhaps relevant...
The focal hyperbola of the ellipsoid and the focal ellipse of the hyperboloid of two sheets, are umbilicar focals, and pass through the umbilics of these surfaces.
In short, there's no grammatical reason why OP can't use umbilicality in his context, but obviously it makes sense to use whatever word his colleagues use. A quick search on Google Books claims 96 instances of umbilicality and 305 for umbilicity (plus 3 for umbilicalness). As I said, I'm no expert, but it looks to me as if they're all intended to mean much the same thing. I'd go with the majority.
Umbilicality is a word, in the OP's context of mathematics and in mathematical physics, used to form a noun expressing state or condition from the adjective umbilical.
Levi umbilicality is weaker than Euclidean umbilicality because it contains no information on terms of the form h(Z,W) with holomorphic Z and W. [p.1]
Concerning the restriction n>=2 in the classification theorem, note that for n=1 the umbilicality property is satisfied by any hypersurface of C2. [p.2]
R. Monti, D. Morbidelli: Levi umbilical surfaces in complex space, 01/2006, ResearchGate [Arxiv pdf 243 KB]