Time-Sensitive Echo

CJam, 45 41 39 36 34 bytes

{eslN1$}g;es](es-fm3/{){_es>}g;o}/

This doesn't really make sense in the online interpreter of course, but it works in the Java interpreter.

It doesn't display a prompt.

Explanation

{        e# Do while... (popping the condition from the stack)
  es     e#   Get the current timestamp.
  l      e#   Wait for a line to be entered and read it.
  N      e#   Push a linefeed.
  1$     e#   Copy the line we read - this terminates if the line is empty, because
         e#   empty strings/arrays are falsy.
}g
;        e# Discard the last linefeed (the one after the empty input).
es       e# Push the current timestamp (corresponding to the last, empty, input).
]        e# Wrap everything in an array. This is now a flat array containing:
         e#   - The initial timestamp.
         e#   - Three elements for each line: the line, a linefeed, the timestamp.
         e#   - Two elements for the last line: the empty string and the timestamp.
(        e# Pull off the initial time.
es-      e# Subtract the current time, which gives (minus) the difference between
         e# when a line was entered and when it should be printed back.
fm       e# This maps "minus that value" onto each element in the array. Now the lines
         e# and linefeeds are strings (arrays) - so minus is set difference, but they
         e# only contain characters, not any integers (like the difference value), so
         e# none of the strings will be affected.
         e# The timestamps on the other hand will be incremented by the time difference
         e# between reading and printing, giving the time at which each line should be
         e# printed back.
3/       e# Split the array into chunks of 3 (where the remaining two elements are
         e# just grouped as a pair).
{        e# For each of those chunks...
  )      e#   Pull off the timestamp.
  {      e#   Do while... (popping the condition from the stack)
    _    e#     Duplicate the target time.
    es>  e#     Check if it's still greater than the current time.
  }g
  ;o     e# Discard the target time and print the rest of the current chunk, which will
         e# automatically be flattened/concatenated into a single string.
}/

JavaScript, 119 112 bytes

k=(d=Date.now)(i=j=[]);do{i[++j]=[prompt(),d()-k]}while(i[j][0]);i.map(a=>setTimeout(b=>console.log(a[0]),a[1]))

Hoping to find a couple more bytes to cut out.

JavaScript, 120 bytes

No chance of getting near to CJam with this approach, but a straightforward script.

a=[];t=+new Date;while(s=prompt()){a.push({s:s,t:+new Date})}while(v=a.pop()){setTimeout(`console.log('${v.s}')`,v.t-t)}