The FCO, as a matter of successive government policy and international norm, offers consular assistance to all and any British nationals arrested abroad. I think this is entirely right.
That assistance means things like ensuring you are being treated fairly in accordance with the laws of the country, that you can understand what is hapoeneing (they can signpost, but do not pay for reliable lawyers and interpreters) and ensure that the conditions under which you are held are not worse than those for other prisoners. They will also make a stand against any form of prisoner abuse (inhumane ot degrading treatment, denying medical treatment etc).
They may also get involved if your trial is not help in a timely fashion, or there is no prospect of a trial.
They will also check your eligibility for transfer to prison in UK if there is an agreement with that country to move prisoners. And yes, that includes smugglers, rapists, murderers
They will also protest against any death sentence, even when handed down correctly in that country, and make a plea for clemency and commuting the sentence to life imprisonment (many countries which do not have the death penalty do this for their nationals convicted abroad). It may well not work, but it is always attempted.
They do not however seek to overturn what appear to be sound convictions for criminal offences. They may get involved with apparent 'political' offences and miscarriages of justice.
If this family feel that the conviction was unsound, and have evidence that she did not carry in the drugs, then of course the FCO should be involved. I do not however thing that is what has happened here. Her sentence is 3 years (maximum 25 years or, if with aggravating features, death) which is by local terms lenient. And Egyptian prisons are not of the standard of UK ones, but I've not heard that she is in worse conditions than the country norm. Does anyone know if that is the case?
FCO role now should be to see if transfer to serve her sentence in UK can be secured, and as quickly as possible.