Speaking from experience, I can say this transition is doable, rewarding in every way, and frankly I don't know why this is even still a question today.
Anyone saying Linux or open-source software "isn't safe" has no idea how networking works -- they're certainly already using several Linux-based devices and servers in city buildings right now. Linux and UNIX machines vastly outnumber Windows installations today; you can't use the Internet without interacting with hundreds of Linux machines per hour.
Next time someone tells you they don't want to use Linux, UNIX, or open-source software, remind them that their smartphone runs one of those two operating systems. And yeah, no more Internet.
I've had some success getting around prejudices by explaining to people what the RFCs are, and how those are the Internet, and the relationship between open standards and open software, but that's a longer discussion.
Having said all this, and institutional inertia being what it is, I think you'll have an easier path advocating a switch from Windows to Chromebooks or ChromeOS workstations for the average user, with Ubuntu for more complex workstations only. If you really want Ubuntu everywhere, then look at techniques that are ChromeOS-like in management: containerize as much as you can, use snap, look at management tools such as Chef. Other than virus prevention, the real win in institutional Linux is the lower TCO and higher reliability from automated administration.
Either way, step one is to set up a Google Apps domain for the city, and get people started using google docs, deprecating Word and Excel. You'll see immediate productivity gains just from the collaborative editing capabilities. If you're in a situation where this can't be done for legal or firewall reasons, then it might not be time yet for you to push the migration. You might be able to accomplish similar "cloud-like" results with internal servers and something like ownCloud.
In California, Chromebooks in each classroom have just about supplanted the Windows-based computer lab in most schools. This provides a more secure, more manageable platform with a much lower total cost of ownership; so low that schools are approaching a 1:1 ratio of systems to students, where before they could only afford around 1:20.
Another data point: The New York financial industry figured out a couple decades ago that Linux TCO was a fraction of the cost of Windows. Even UNIX, with its higher license fees, was cheaper, since most of TCO is labor. I used to build trading floors; by the time I left NYC, we had 5 guys managing thousands of UNIX machines with primitive tools we wrote ourselves. The ratios just don't favor Windows, and never have.
Google around for Gartner Group and other consulting agencies' reports on Linux TCO -- you should find more stories like this there.
A final data point: among other things these days I'm the "IT guy" for our family business. We have around a couple dozen linux machines and one Windows machine. That Windows machine consumes about half of my sysadmin time, and the only thing it runs is quickbooks. Not cost effective at all.
Windows started life as a personal computer UI, optimized for home use by a single user, and even after gaining its own kernel it's always struggled to grow out of that mindset; the Windows development community is still stuck there. In terms of security, it's a toy. In terms of institutional use, it's completely inappropriate.