The Groove music app on Window phones do not support scrobbling to Last.FM, and the company of course does not have a native app for the OS.
There are a variety of 3rd party apps on Windows Phone which lets you scrobble your tracks to Last.FM, but recent back-end changes by Last.FM has rendered al of them inoperable.
Fortunately one such app, ScrobbleMe, has seen a rapid update which has restored the features, with the developer also saying they are planning a Windows 10 Mobile specific version of the app.
Besides letting you scrobble your recently played songs to Last.fm with just one tap, you can also synchronize your loved songs with your Last.fm profile and view your recommended artists, top artists, recently scrobbled and loved songs.
See screen shots of the app below.