You are not required to press TOGA to get into WS mode. Only if you are on an ILS. This is from the FCOM. In reality it matters very little as we press TOGA as part of the manouver.
Non-normal procedures ( Boeing Standard)
Windshear Escape Maneuver
Pilot Flying
Pilot Monitoring
MANUAL FLIGHT
• Disconnect autopilot.
• Press either TO/GA switch.
• Aggressively apply maximum*
thrust.
• Disconnect autothrottle.
• Simultaneously roll wings level and rotate toward an initial pitch attitude of 15 °.
• Retract speedbrakes.
• Follow flight director TO/GA guidance (if available).
• Verify maximum* thrust.
• Verify all required actions have been completed and call out any omissions.
AUTOMATIC FLIGHT
• Press either TO/GA switch**.
• Verify TO/GA mode annunciation. • Verify GA thrust.
• Retract speedbrakes.
• Monitor system performance***.
• Verify GA* thrust.
• Verify all required actions have been completed and call out any omissions.
MANUAL OR AUTOMATIC FLIGHT
• Do not change flap or gear configuration until windshear is no longer a factor.
• Monitor vertical speed and altitude.
• Do not attempt to regain lost
airspeed until windshear is no longer a factor.
• Monitor vertical speed and altitude.
• Call out any trend toward terrain contact, descending flight path, or significant airspeed changes.