You are presuming that he's using the connection of the main Ecuadoran embassy (for which it may have more than one). And, yes doing so would likely cause a diplomatic issue, However I'd like to propose a couple of things :
* It does not say which state. If the connection is a separate one facilitated by the Embassy, it does not say that Ecuador did not cut it.
* If could be a separate connection not used for official business by the Embassy, but for visitors etc. but on contract facilitated by the Embassy that was then cut by a request to the provider by the UK.
There are a number of ways this could be setup, and still be consistent with the published statement. Perhaps notching back on the histrionics would provide, overall, a better perspective on whats actually in play here.