This is an topic that recently came up in book club. 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 by the students in the class and these sounded like good suggestions but 'lately' didn't sound quite right.
My dictionary phone app didn't particularly help 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'!