After a bit of searching it was clear that the font used in Writer wasn’t available but there was a free alternative as part of Google Fonts called Cousine. The rest of this post will show you how you can get this font onto your iOS device. Firstly download the font package from the Google site.
You will need a copy of a font installation app. There are a number available but the one I am using is InstaFont which costs £1.49. You will also need a way of getting the font onto the device. I used Dropbox but anything that can allow you to download the file and open it in another app will do so you should be able to use Google Drive/OneDrive/Box or a Mail app.
Once transferred you should see the font appear in your custom font list. Select it and the install button in the top right will show (1). You can then click this to move on to the installation.
When you press the Install button the app will switch to Safari and them almost immediately to the settings app where you will then see the Install Profile screen:
You will then see a warning message about the profile being unsigned. You can safely ignore this and press Install:
For some reason pressing Install twice isn’t enough for Apple so go ahead and press it once again:
The installation is now complete and so you can press the Done button:
The font is now installed and ready to use in other apps. For example here it is being selected for use in Drafts:
And finally here it is in use. You can compare with the original in iA Writer immediately below it. Pretty similar I think.