Missed approaches or go around are frequently mishandled, the most common being delayed gear retraction, no big deal on what will have been a fairly light aircraft with both engines operating.
If if there was wind shear even without a warning I would delay gear/ flap retraction as the wind shear guidance computer gets messed up by configuration change.
TOGA should always be pressed other wise advancing the thrust alone will simply accelerate the aircraft towards the ground as it could still in in G/S mode, one press of TOGA ( TAKE OFF/GO AROUND) will give you a FD pitch up and thrust guidance and change from G/S and normally a 1000-2000 fpm rate of climb using reduced go around thrust, a second press will give you full G/A thrust and a ballistic rate of climb in a light and very powerful aircraft.
For wind shear escape after a wind shear warning (not caution) its all about terrain avoidance and advancing the thrust levers to full thrust ( FDEC prevents exceeding thrust limits) and pitching to the PLI’s ( pitch limit indicators)
In this go around and given the forecast wx covered an extended period I too would have diverted to my alternate, yes they might have got in on a second attempt but why push it? go to MAN refuel and put the passengers on a MAH LHR shuttle if need be.