According to the Oxford English Dictionary, *directly* can be used colloquially as a conjunction:
> colloq. as conj. As soon as, the moment after. (Elliptical for directly that, as, or when.)
The examples they give include:
>1795 Montford Castle I. 88 Directly you refused [his] assistance, a judgement overtook you.
>1827 R. H. Froude Remains (1838) I. 68, I quite forget all my scepticism directly I fancy myself the object of their perception.
There is some discussion of this conjunction [here](https://books.google.com/books?id=_QISAAAAIAAJ&pg=PA71&lpg=PA71&dq=directly+conjunction+colloquial&source=bl&ots=NFhxoBR9qg&sig=YfPA3UUTE79QIOlbhAgYoqhjB2E&hl=en&ei=oLdeTrKiE8nHsQKRsaEJ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEcQ6AEwBg#v=onepage&q=directly%20conjunction%20colloquial&f=false), which noted:
> ![enter image description here][1]
So this use of *directly* is a Britishism. Similar words are *immediately* and *instantly*. Not all adverbs function as conjunctions, however, and there was debate about how proper such a use is (Dickens used them, but dictionaries didn't like the construction--note that the OED still calls it colloquial). The book mentioned above also notes that it is possible that any adverb could become a conjunction, so future uses of adverbs cum new conjunctions could occur:
> 
The book notes that *once* and *directly* were being adopted at time of writing. Only *once* seems to have taken hold ("I'll be with you once I do XYZ") while *directly* remained a mostly British usage.
On formality, the Cambridge Dictionaries Online note that the phrase is [slightly formal or formal](http://dictionary.cambridge.org/dictionary/british/directly_4), depending on how the conjunction is used.
[1]: https://i.sstatic.net/3ks8u.png