How did Arkangel still work after Sara became a teenager?

There's no real indication that it requires a backend.
Maybe we're supposed to believe it only needs the implant & 'iPad' to function.

This is a similar theme throughout Black Mirror. Technologies are simply a "given" with little explanation as to how they function. Historically, much good sci-fi has done the same. You don't need to know how it works, or even its history, to appreciate the tale being told using that tech as a backdrop. You just have to accept "it works".

Many of the Black Mirror stories are actually 'moral tales' - they just use a tech backdrop to provoke a stronger sense of it not being quite current-day, even though the audience is aware it is indeed a 'mirror' of current society... hence the title.