The upgrade from Mint 18.3 to 19 is described here. https://blog.linuxmint.com/?p=3615 if you follow it exactly, once it is done, if you go to the update manager and edit it will show the upgrade to 19. It will then go through the process for you. Once that is done you can go from there to the same place, which will lead to the 19.1 upgrade. There is a problem with some updates on 19.1 but these are shown as broken packages, which are fixed in Synaptic Package Manager which is easy to do do as it shows broken packages there. Then you reload the packages and it set things right. If you then go back to Update Manager and refresh it will then reset things and complete. Since I wrote this I see the update problem in Mint 19.1 has been resolved. You may still have to refresh during it, but that is all