Period in which one’s at home drinking tea; Alice never quite made it (5,5)
I believe the answer is:
daisy chain
'alice never quite made it' is the definition.
(reference to Alice in Wonderland)
'period in which one's at' is the wordplay.
'period' becomes 'day' (day is a kind of period).
'in which' means one lot of letters goes inside another.
'one's' becomes 'I's' (Roman numeral).
'at' becomes 'chain' (I am not sure about this - if you are sure you should believe this answer much more).
'day' enclosing 'is' is 'daisy'.
'daisy'+'chain'='DAISY-CHAIN'
'home drinking tea' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
(Other definitions for daisy chain that I've seen before include "Simple garland" , "simple floral arrangement" , "String of flowers" , "Connect IT devices" .)