They have no authority.
They do, however, have the authority to request the foreign country to extradite him or her and the foreign country must then decide whether that extradition can proceed.
The foreign country will refuse the request either because it's politically expedient to refuse or, alternatively, it's not possible for them to extradite for the particular reason due to their laws. On the whole, countries are sensible and do not request extradition unless there's a reasonable chance the extradition will be approved. Hence Saudi Arabia doesn't bombard the US with ridiculous extradition requests for Saudi citizens drinking alcohol in public in the US. (Or US citizens for that matter)
Were Assange to be arrested by the British Police on leaving the embassy, the UK would, for example, refuse to extradite him unless they get a guarantee from the US that he will not face the death penalty. That is irrespective of any crime he might have committed or whether that is an extraditable offence.
(In this particular case there would also be an extradition request from Sweden. I have no idea how the UK would prioritise them but I would expect that they would not extradite to Sweden unless they also guaranteed to require that the US guarantees no death penalty on a subsequent extradition.
Interestingly, if Assange could not be extradited to the US legally from the UK then the UK would (probably) refuse to extradite him to Sweden unless they guaranteed that he wouldn't be extradited to the US.
But as I suspect it's easier to extradite from the UK to the US than from Sweden to the US, I would expect that the US would try to get priority over an extradition from the UK.