Is a full timeout always a full timeout ?

[WeekendRef]

Had a situation this weekend that I was a little unclear about and couldn't find anything in the book about .
Team A is down by 30 late in the second half and calls a full timeout . After about 20 seconds Coach of team A sends his kids back on the floor and starts yelling at us (2 man crew) that he is ready and we should put the ball in play . Team B is still in the huddle under the impression that they have 60 seconds. Coach of Team A really starts to vocalize that we should put the ball in play because he called the timeout and he is ready to play and therefore we should put the ball in even though Team B is not on the floor . I ask Team B if he is ready and thankfully he puts his team back on the floor immediately .
My long winded question is this : Does the team that called the timeout have the ability to determine the duration of the timeout ? Should I have told Team A to cool their jets and alotted Team B the full 60 seconds ?
I asked a couple of guys that I have worked with and I got a couple of different responses...