The FAA AIM and ATC (7110.65) documents don't mention the ICAO equivalent.
But I found it in the US AIP : 34.1
(...)
At times a clearance may include the word “IMMEDIATE.”
For example: “ CLEARED FOR IMMEDIATE TAKEOFF. ”
In such cases “IMMEDIATE” is used for purposes of air traffic separation.
It is up to the pilot to refuse the clearance if, in the pilot’s opinion, compliance would adversely affect the operation.
aviation.stackexchange.com/users/14897/ymb1
Assuming US/FAA: The Pilot/Controller Glossary only has one instance of "no delay": MINIMUM FUEL − Indicates that an aircraft’s fuel supply has reached a state where, upon reaching the destination, it can accept little or no delay.
This is not an emergency situation but merely indicates an emergency situation is possible should any undue delay occur.
"Immediate takeoff" doesn't exist, either, but we do get IMMEDIATELY − Used by ATC or pilots when such action compliance is required to avoid an imminent situation.
We can cross-reference this with the Skybrary entry for Immediate Takeoff :
When given the instruction ‘cleared for immediate takeoff’, the pilot is expected to act as follows: At the holding point: taxi immediately on to the runway and begin a rolling take off without stopping the aircraft.
If it is not possible to begin taxiing onto the runway at once or if take off performance calculations mean that a standing start is necessary, then the clearance must be declined If already lined-up on the runway: commence take-off without any delay.
If this is not possible for any reason, the pilot must advise the controller immediately.
So that suggests that "immediate takeoff" is more standard.
My experience has always had the clearance be "no delays".
If the P/CG doesn't technically define either phrase then I can suppose that they use "no delays" to avoid the knee-jerk "I must do something" to "immediately".
For "no delays" you can always easily respond "unable" and wait until the next plane lands.
aviation.stackexchange.com/users/15703/bartonjs
According to the ICAO Doc 4444
14th ed.
(Procedures forAir Navigation Services - Air Traffic Management )
They mention a clearance for "immediate take-off" in section 7.8.3.4: 7.8.3.4
In the interest of expediting traffic, a clearancefor immediate take-off may be issued to an aircraft before itenters the runway.
On acceptance of such clearance the aircraftshall taxi out to the runway and take off in one continuousmovement.
I don't see any reference to "take-off, no delay" or "cleared for immediate take-off" in the FAA ORDER
JO 7110.65W (air traffic control procedures and phraseology).
I believe both are correct with ATC opting to use "Cleared for take-off, no delay.
" to shorten their transmission to save time.
aviation.stackexchange.com/users/30939/TK252
There's also the option of expedite .
EXPEDITE− Used by ATC when prompt compliance is required to avoid the development of an imminent situation.
https://www.faa.gov/air_traffic/publications/media/pcg_4-03-14.pdf
aviation.stackexchange.com/users/31010/CaptObvious
Here's one from the field ,
TOWER:
Southwest 3828, Midway tower, runway 31C, line up and wait.
Don't plan on stopping .
SWA3828:
31C line up and wait, Southwest 3828.
TOWER:
Southwest 3828, traffic holding in position on the crossed runway, traffic on 3-mile final for the crossed runway, no delay please .
Turn left heading 250, Runway 31C, cleared for takeoff.
The wind 060 at 9. SWA3828 and DELTA1328: Heterodyne
What follows the heterodyne is what makes this one famous, but the answer to your question as plays out here, is the controller intentionally used informal language in his advisory "don't plan on stopping" and the admonishment "no delay please".
aviation.stackexchange.com/users/22864/Harper - Reinstate Monica
Just wondering - what's the correct ATC phraseology for a takeoff where a plane does not stop first after lining up with the runway centerline before starting their takeoff roll?   Redwood five-two-six-three, runway three-zero cleared for takeoff, no delay.  Redwood five-two-six-three, runway three-zero cleared for immediate takeoff.
What is the correct ATC phraseology in the US for an immediate - no delay takeoff?
aviation.stackexchange.com
<air-traffic-control><usa><takeoff><phraseology>
aviation.stackexchange.com/questions/51456
aviation.stackexchange.com/users/25068/slantalpha
"Immediate", "Expedite" and "minimum fuel" are terms used to indicate a no delay take off. Usually, "IMMEDIATE" or "IMMEDIATELY" is used by ATC or the pilots for air traffic separation. "EXPEDITE" can also be used when quick compliance is vital to avoid an imminent situation. Finally, in the Pilot/Controller Glossary there is only one instance of "no delay":   MINIMUM FUEL   −   for when an aircraft is almost out of fuel and upon arriving, there needs to be no delay.
Usually, "IMMEDIATE" or "IMMEDIATELY" is used by ATC or the pilots for air traffic separation. "EXPEDITE" can also be used when quick compliance is vital to avoid an imminent situation. In the Pilot/Controller Glossary there is only one instance of "no delay":   MINIMUM FUEL   −   for when an aircraft is almost out of fuel and upon arriving, there needs to be no delay.