Translation:Recently my aunt has been going to bed late.
It describes a repetitive action (qualified in time by the "recently/lately"), you could grammatically use either ложится or ложилась, the difference being that with the former, this is something that was happening "recently" and is still happening, whereas if you used the past tense ложилась it would mean that though it did happen recently, it may no longer be the case now. Either one should be correct with this English though. I had a drop-down choice exercise, so it wasn't an issue.
Isn't that the Oxford English Dictionary definition #1 of "recently"?
Anyone else, please chime in, but it seems like the difference in usage is as follows: the first phrase relates to events that occurred not long ago but are still ongoing; the second relates to events that occurred not long ago but are no longer ongoing.
Is that a fair understanding of when to use one phrase versus the other?
The Wiktionary translation for "в последнее время" is "these days," which maybe helps get at the contrast with "recently." Somewhat subtle, but I'd speculate that if you were comparing, say, the 1980s to the 1800s, "недавно" might be a better choice. Of course I'm not a native speaker.