I think that "últimamente" implies some comparison with the previous period, whereas "recientemente" just gives us a time reference, which is recent. Also, "últimamente" could be a shorter interval than "recientemente." Examples:
Últimamente no se ve muy contenta mi hermana. [The implication is that previously she seemed a perkier.]
Ha habido mucha lluvia recientemente. [This doesn't mean necessarily that there was significantly less rain earlier. The sentence just means that recently there was a lot.]
In your examples, I think that últimamente would likely be taken to cover a shorter term than recientemente.
Also, as Pablo said in a comment, "recientemente" can come across as rather formal. However, the adjective "reciente" has less of a formal feel to it, for example:
- ¿Es un cambio muy reciente?
user2325442 came up with a great example (I was in a car accident/Choqué mi coche). This shows that "últimamente" only works for ongoing actions, whereas "recientemente" can in principle work for ongoing or single point in time actions.
Bottom line, for a conversational tone, the following are recommended: