British would not hesitate a single moment before the invasion to Turkey. Just see the timeline. The British were victorious on Middle East theatre in 1941.
May 1941 - Iraq (IOTL)
June 1941 - Syria/Lebanon (French Mandate of Palestine as in 1941) (IOTL)
August 1941 - Iran (IOTL)
October 1941 - Turkey (ATL)
Iraq and Iran :
no tru "war", more or less only civil wars, the Brits overthrowing to them "unsuitable" regimes
Syria/Lebanon/Vichy :
what do you expect from already beaten french, who out of escapisms haven't really surrendered to the "common goal" and rallied against the "common enemy", who already/still leaned towards the Wallies ?
All in all : none of these operations could compare with the scale of operations in other theatre, if german forces are seriously involved.
And for
... not hesitate a single moment before the invasion to Turkey ...
the turkish army was quite a different beast even to the few Vichy forces, not to speak of Iran and Iraq.
Counter-arguement during negotiations :
with "proper" support via turkish bases non of this would have happened.
with "proper" based in turkey all that could be regained ... at least partially also for Turkey to gain