Thank you for reaching out to Microsoft Q&A Platform .
To answer your question , Yes this operation should be seamless and should be in effect rite away .
What basically happens in the background:
Azure stores 3 copies of your data in 3 different places with in the primary location for hot Tier { Active, Secondary, Standalone}
Once you choose cold tier it will remove one copy from above .
There is no additional cost for the same , Of course you will start saving more money after this transition
request you to please do accept the answer if your question was answered .
Thanks & Regards,
Pradeep