There is no one answer here as every family structure is different. Like most other comments here, my 2 (5yo, 22months) children have first dibs on my time while I am not on the clock or am not dealing with a production issue off hours. Once they go to bed (sometime between 7:30-8:30pm), I determine if I have enough in the tank for either: 1) Learning 2) Spending time with my wife (second dibs on my time).

In order to "stay relevant", I have found this to work for me: 1) During my passive commutes (every morning and afternoon), I have a solid 30 minutes of either listening to a podcast if I don't get a seat or toying around with new languages/frameworks otherwise. 2) Convince your current company that it is in their best interest to bake learning into your current job.

Obviously (2) can be quite difficult, but it starts a conversation you want to have anyway about making sure the entire team stays relevant and fresh. Most people always point to Google's innovation time as a good measure of how this should be done.

Having children has absolutely changed my priorities around and put an increasing value on my time. For me, I would much rather forgo learning the new JS flavor of the week in favor of spending as much time as possible with my growing children.