Although there are some elements of a shared meaning, they are usually used in context very differently.
"exactly" is used as both a reaction response for confirming and as part of a full sentence for wanting more detail.
---> someone asks "so I ALWAYS have to tell my boss when I want to leave work?" and the response "exactly".
---> someone asks "what exactly do you do for work?".
"spot on" is definitely very British, you're right about that, and is used as a reaction response to a good job having been done, indicting accuracy and to confirm.
---> someone says "I got all of the paperwork filed, so we can go home early" and the response "spot on"
---> someone says "your report is spot on"
---> someone asks "so I need to be at the interview 30 minutes before it starts?" and the response "spot on".
Both examples of "exactly" are very commonly used but regarding "spot on", it is only used in the UK and mostly for the denoting accuracy (the 2nd example).
I hope this helps.