This is a topic that came up in book club a while ago and today in conversation class. Someone in the class said 'shortly I saw a series about...'. I highlighted that 'shortly' wasn't the appropriate word in this sentence and we looked for alternatives. Both 'recently' and 'lately' were suggested and these sounded like good options but 'lately' didn't sound quite right.
My dictionary phone app wasn't particularly helpful as it says that 'lately' means 'recently'. However, the example sentences did shed a bit of light on it, as did a quick Google search (using the terms 'recently' 'lately' and Cambridge).
We typically use 'lately' with the present perfect whereas recently can be used with both the present perfect and with the past simple.
I recently saw a series about...
Lately, I've been watching a series about...
Have you done anything interesting lately?
Have you done anything interesting recently?
Finally, if in doubt, use 'recently'!