As this is a Skype for Xbox One forum, I'd check two things first:
1. Open Skype on Xbox One, hit Menu button to open Settings. Make sure notifications are turned on there.
2. When at your Xbox One home screen, hit menu button and go to Settings > Preferences. Make sure notifications are turned on.
I have a similar issue and it requires you to turn on notifiaction. XB1 will take a little longer to notify (popup at the bottom of the screen) you compared to other devices. For example, my phone will notify on an incoming call, then my tablet and lastly the XB1. You should expect someone to wait at least 15-30 seconds. I use Skype often on my XB1 and all my friends are used to waiting for calls to connect since it's that 30 seconds & time for you to respond. If skype isnt already running, it'll take about a min to load and then 30+ seconds to connect so expect on the short end, 2 min.