TL;DR Remove all music from your iOS device and re-sync from iTunes for custom lyrics to show.
After upgrading to iOS 10 I noticed that none of my custom lyrics were showing for any items in my library. Posts such as http://www.mcelhearn.com/no-lyrics-for-music-in-the-ios-10-music-app/ seemed to suggest that lyrics were only available for Apple Music subscribers and if that was you you'd still need to be online as they'd be fetched as each song plays.
A few tweets back and forth between @AppleSupport and they agreed that it was restricted to Apple Music subscribers and sent me over to https://support.apple.com/en-us/HT204459. Except there it clearly states that you're able to add custom lyrics in iTunes and if iTunes is showing it then it should be available on iOS. I got a case number from Apple and while on the line to an agent I noticed that some tracks were indeed showing lyrics. Odd. Thinking I'd made a mistake I told the agent that it seemed to be working now and left it at that.
Except this morning I noticed that some tracks were still without lyrics. Turns out the ones that were working had recently been sync'd across from iTunes and sure enough removing all music and re-adding sorted the issue. I've let Apple Support know so hopefully if others contact 'em they won't be fobbed off like I initially was.