There are quite possibly other diferences that matter in a discussion of the equivalence of the pairs of terms in the question; the answer should not be taken as the last word on the subject.

1. The forms “tellement de” and “tant de” are equivalent (same meaning); the equivalence is however not perfect; they are both used with a word called a “correlative” in French and that word, “que”, is a consequence of their meaning (1) (2): they are used to say that a state of affairs has reached a high (low) enough level , for a subsequent condition to be fufilled or for a subsequent action to take place; it is the word “que” which is used to introduce that subsequent action, state or quality; they have the same register, which is formal and current speech; one différence between the two is the domain of application; both are used for the high end; for the low end only “tellement” is used with an added adverb, “peu“. examples.

high

Il a tellement/tant de voitures qu’ il a oublié posséder une Fiat.
Il y a tellement/tant de monde dans la rue que l’on ne peut plus avancer.
Ce travail de recherche demande tellement/tant d’argent que seulement quelques pays s’en occupent.

low (“tant” ne peut pas être utilisé)

Les ruisseaux ont tellement peu d’eau que beaucoup de poissons meurent.
Les fruits ont tellement peu de gout que les gens ne les achètent pas.
Elle a tellement peu de temps que tous ses dimanches sont passés au travail.

2. “énormément de” and “beaucoup de” are not informal locutions and they are equivalent
   insofar as they indicate large quantities; they are not equivalent on the count of their respective indication of the degree of largeness: the first is used for quantities that are exceedingly great by normal standards; the second is used for quantities that are considered to be reasonably greater than what one usually would think to be normal.
   The speech register of both is the same: they can be used in every day speech and in writing; there is nothing informal about them.

3. “Tellement” is a word that has evolved since the XIIe century and I really can’t see what colloquialism would have to do with it, specially as it is a term with a very simple and important function in the language. May I ask you, though, just in case, if you were considering a link with “to tell”? (It has nothing to do with “tellement”.)

I’d say that tant is slightly more « relevé » than tellement, and tellement peu and si peu are equivalent (« Les fruits ont si peu de goût que les gens ne les achètent pas »).

Vous pouvez dire aussi : « Il a un nombre si grand de voitures qu’il a oublié qu’il possédait une Fiat » (on le comprend …)

Honestly nowadays it will shock very few people if you employ those expressions in formal French language, they are perfectly valid in my opinion.

The only thing is that’s they might exaggerate things a little bit more than ‘tant de’ and ‘beaucoup de’, but if you’re careful, no problem.