The only mention of the Unseen Servant spell ending due to distance from the caster is found at the end of its description:
If you command the servant to perform a task that would move it more than 60 feet away from you, the spell ends.
Compare this to a similar, but lower-level spell, Mage Hand, which includes the following in its description:
The hand vanishes if it is ever more than 30 feet away from you
Say an unseen servant is spawned in a location where it can complete a task without moving. Will the spell end if the caster commands it to do the task while being more than 60 feet away from the servant?
Similarly, say an unseen servant is spawned in a location where it can complete a task without moving. The caster, while within 60 feet of it, commands it to do the task at a predetermined time in the future (e.g. "1 minute from now", or "once I cross this bridge"). If the caster is then 60 feet or more away when the time comes to do the task, will the spell end?
Unseen Servant's spell description of what it is capable of doing and how it does it:
The servant can perform simple tasks that a human servant could do, such as fetching things, cleaning, mending, folding clothes, Lighting fires, serving food, and pouring wine. Once you give the command, the servant performs the task to the best of its ability until it completes the task, then waits for your next command.
In both scenarios, I think the Unseen Servant is able to complete the task without the spell ending. My reasoning for this is the specific wording of "that would move it more than 60 feet away from you", a wording not found, as far as I know, in many (any?) other spells, & the glaring lack of anything along the lines of the quoted portion of Mage Hand's description, a version of which is found in numerous spell descriptions.
In the second scenario specifically, I think the task is completed without the spell ending for an additional reason, which is that the task starts once the command is given & waiting for the appropriate trigger is, mechanically, the first part of a two-part task.
I would really appreciate hearing the community's thoughts on this one.
I did read a similar Unseen Servant question which has offered some insight:
Does the Unseen Servant spell end if I move away from it?
However, it doesn't quite cover everything I've put forward. The three scenarios presented in the other question do not cover the same mechanics as mine. Receiving a command from more than 60 feet away - theirs involves the Servant moving; mine does not. Having the caster move more than 60 feet away away after issuing a command is only mechanically similar if the task is, as I laid out in my example, a 'two-part task'. Certainly, the two questions are similar, but the specifics presented are not the same.