Well Luda are you suitably confused?
First comment is please do not think all RT examiners have mastered CAP 413, they havnt!
You may want to consider
Student flights
2.33 On initial contact, student pilots who are flying solo
shall use the callsign
prefix ‘STUDENT’.
Flying instructors should be well aware of this call, especially those who claim to understand duty of care!!
As 99% of people I examine have not got a PPL(which I would imagine you havnt) I expect them to use the above. It also proves they have read CAP 413 which clearly some of tbe respondents to your post havnt!!
If you treat the test as a practical exercise why would you just request a diversion to Odiham because of weather? Your first call in fact would surely be a request for the weather at your chosen destination, little point diverting to an airfield where the weather conditions may be worse than your original destination. Using the controllers knowledge and information is also an important part of CRM. The controller will most likely have a much better idea of suitable alternatives weather wise than you will.
You should also really know how to temporarily leave the frequency and obtain weather from the FIR frequency(not part of the RT test)
If you are receiving a LARS service your diversion call would need to include the word 'request' if you were inside controlled airspace but is technically not required if you are in Class G airspace but it would be advisable to check with the controller before making large heading changes.
How would your alternate know you were coming on first call?
Your first call should make that clear
3.28 Normally, the initial call to an ATS unit should only include the minimum
information needed to establish:
1. The service that an enroute flight requires; or
2.
The clearance/information that a joining or departing flight requires.
But the exact detail of that call would be dependent of what type of airspace you were going to have to transit/penetrate to get to the alternate.
"Request join" may be suitable at Popham but might not go down so well at Manchester International!
Every possible call is not contained within CAP413 so some common sense is needed too, unfortunately that can be difficult to find on the internet sometimes.
Forget garbage such as "thanks for your help" - you are supposed to be demonstrating accurate communication.
Quite correct and instructors, especially, should not be encouraging this sort of RT
You should also read this before your test
https://www.caa.co.uk/docs/33/20130121SSL08.pdf although some of it is out of date
I should also add that a weather diversion isnt necessarily an emergency, it could be just that the crosswind was out of limits. Select a diversion that is appropriate to your situation, a busy military airfield may not be best pleased to see you as a result of a non emergency diversion.