I'm been wondering about this for a while and hope someone can enlighten me.

In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I read the book of Coddington and Levinson on ordinary differential equations and was taken with the topic of differential equations with irregular singular points. There are traces of this topic in the long letter to Weil about the Hecke theory. The same topic reappears in the “geometric Langlands
programme” over C, where it has quite a different flavour than in Coddington-Levinson, whose references will have been Poincare and G. D. Birkhoff. In its current guise it is little explored. I hope to find time to learn more about this before I am done."

From having seen Edward Frenkel speak on dualities in physics and geometric Langlands, and from speaking to people far more familiar with all of the above topics than myself, I can guess that the link with geometric Langlands may be as follows:

ODEs with irregular singular points $\longleftrightarrow$
asymptotic series/resummation methods $\longleftrightarrow$
getting non-perturbative physical theories from perturbative ones $\longleftrightarrow$
the link with geometric Langlands, with the last correspondence as described say here https://arxiv.org/abs/0906.2747.
Is this accurate? Are there any insightful comments one would like to add?

Even if the above has some semblance of truth, I am not sure about what the link is to the arithmetic Langlands program: "There are traces of this topic in the long letter to Weil about Hecke theory."

The second last sentence is fascinating - I have talked to other people who also think "In its current guise it is little explored" (which came as a big surprise to me - ODEs with singularities?!), but to think understanding this better might lead to insights into both versions of the Langlands program (could it?) sounds very interesting.

P.S. That I shared the arxiv paper above should not be taken as an indication that I understand its contents with any amount of depth at all, though if you feel like assuming familiarity with technicalities in the geometric Langlands program will lead to a full answer, please go ahead (and provide references if you can!)