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Operation Sealion
An Axis invasion of the UK will lead to some increased US and Soviet mobilization.
On the other hand, the capture of London or Manchester can trigger Spain and Turkey to lean further towards
the Axis, offering up the possibility of new Axis minors entering the war if some diplomacy is used.
The British will not surrender when London and Manchester have fallen, but will instead have the opportunity
to relocate their capital to either Australia or Canada.

From the manual

In 12.02, this (Canada/Australia option and Spain Mobilises for War) also seems to happen if Cambridge and Maidstone but not London or Manchester fall

That's correct. Spain mobilization can increase if there is an axis unit within 2 hexes of London or Manchester.

; Axis land units in England resulting in a
; 30-40% increase in Spanish mobilization towards Axis:
{
#NAME= Sealion (Spain->Axis)
#POPUP= <<TAG_35>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#GV= 1[1,100]
#LINK= 0[0]
#LEVEL= 0
#COUNTRY_ID= 99
#TRIGGER= 10
#DATE= 1939/09/01
; 30-40% mobilization increase towards Axis
#MOBILIZATION= [30,40] [1]
; Set variable conditions:
; 1st Line - Germany politically aligned with Axis and not surrendered
#VARIABLE_CONDITION= 45 [1] [100] [0]
; Axis land 1 unit in the UK within 2 hexes of London OR
#CONDITION_POSITION= 87,37 [2,2] [1,1] [1] [0]
; Axis land 1 unit in the UK within 2 hexes of Manchester
#CONDITION_POSITION= 85,33 [2,2] [1,1] [1] [0]
}

Moving British capital is essentially triggered when any Axis unit in UK (within 10 hexes of Manchester):

; Event triggered by Operation Sealion
{
#NAME= DE 106 - UK: Shall We Move Our Capital To Australia Or Canada? (AI - 50% Yes)
#POPUP=
#MESSAGE= <<TAG_20>>
#IMAGE=
#SOUND= teletype_1.ogg, teletype_2.ogg
#FLAG= 1
#TYPE= 0
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 112
#TRIGGER= 100
#DISPLAY_ORDER= 1
; Set decision value
#DECISION= 106
#MPP_UPDOWN= 0
#MPP_TURNS= 0
#MPP_TEXT=
#NOTES= <<TAG_334>>
#NOTES_POSITION=
; Set AI acceptance % (AI will accept 50% of the time)
#AI_RESPONSE= 50
#AI_RESPONSE_POPUP= <<TAG_21>>
#DATE= 1939/09/01
#TEXT_RGB= 0,0,0
#SHADOW_RGB= 228,221,198
; Toronto is in Allied hands
#ALIGNMENT_POSITION= 17,47 [2]
; Melbourne is in Allied hands
#ALIGNMENT_POSITION= 217,119 [2]
;Set National Morale Trigger (dummy value)
#NATIONAL_MORALE_TRIGGER= 0 [0]
; Set variable conditions:
; 1st Line - Canada politically aligned with Allies and not surrendered
; 1st Line - Australia politically aligned with Allies and not surrendered
#VARIABLE_CONDITION= 21 [2] [100] [0]
#VARIABLE_CONDITION= 6 [2] [100] [0]
; The Axis have a unit within 10 hexes of Manchester
#CONDITION_POSITION= 85,33 [10,10] [1,1] [1] [0]
}

And it seems if next turn Axis take another town (Norwich) then Turkey lurches as well

bloomin' flip'

Turkey is the same check as Spain (axis unit within 2 hexes of London or Manchester). But both have only a 10% chance per turn of firing which is why they appear kind of random.

; Axis land units in England resulting in a
; 45-55% increase in Turkish mobilization towards Axis:
{
#NAME= Sealion (Turkey->Axis)
#POPUP= <<TAG_44>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#GV= 1[1,100]
#LINK= 0[0]
#LEVEL= 0
#COUNTRY_ID= 111
#TRIGGER= 10
#DATE= 1939/09/01
; 45-55% mobilization increase towards Axis
#MOBILIZATION= [45,55] [1]
; Set variable conditions:
; 1st Line - Germany politically aligned with Axis and not surrendered
#VARIABLE_CONDITION= 45 [1] [100] [0]
; Axis land 1 unit within 2 hexes of London OR
#CONDITION_POSITION= 87,37 [2,2] [1,1] [1] [0]
; Axis land 1 unit within 2 hexes of Manchester
#CONDITION_POSITION= 85,33 [2,2] [1,1] [1] [0]
}