- Learn to use the tool(s) they provide
- When you learn all there is to know, chances are you will find significant friction, so use your old workflow
- Be ready to bring out the "standard" workflow when getting/giving assistance.
I note that none of these answers are from the perspective of a Vim power user*
*I define Vim power user as anyone who runs into problems with various plugins because they do not conform to <choose your vim behavior>.
On behalf of power (Vim/Tmux)
Vim and tmux make for an insanely powerful combination. If you don't yet have the binding setup, adding something like this:
let session = input('tmux target session:pane> ', ':1.1')
let command = input('test command> ', 'py.test --cov')
"let global = input('bind for all windows? ', 'y')
execute "nnoremap <cr> :w!<cr>:!tmux send-keys -t " . session . " \"" . command . "\" C-m<cr><cr>"
nnoremap <leader>st :call BindTest()<cr>
Allows you to hit Enter to run your automated tests in your split pane, and while they're running you can keep editing. Obviously change the commands to suit your preferences/environment. I use this as an example, but there are many, many other extremely powerful things that you can do with vi/vim/tmux that do not require you to take your hands from the typing position. Any other workflow that I have tried has always left me wanting for the good ol' fashioned command line.
In general, if there is a tool that you have available such as vim+tmux you should use it. I do not believe that it's possible for someone in their entire life to learn and take advantage of everything that they can about vim+tmux.
I suspect that if you feel passionately enough about a non-standard tool that you want to use, it's probably something that has this kind of power.
On behalf of standards
On the other hand, are you so sure that your tool is as good as you think it is? It might not be. The only way you'll know for sure is if you try the other thing. There are some pretty neat features of Sublime/Atom/Brackets/ that can make your life better. Or at least maybe compete with your tool.
Learn the tool that your company provides/requires. Learn to use its shortcuts/hotkeys, learn its features. Is it faster to just use the keyboard? Or can you operate it more quickly with mouse and keyboard? If your initial impression is right, that your tool is better than the standard one, chances are within a week or two you will have explored pretty much everything you can do without extensive customization (e.g. plugins or programming your own extensions). If you aren't continually learning new features that can improve your speed, and especially if you're facing friction in your workflow, now you can put this tool back on the shelf.
It's nothing personal
Most tools that are designed for mass consumption. (<insert tool that you're balking at here>) is designed to work well for the lowest common denominator. I'm pretty sure that I could plunk any of my children down in front of Sublime and they would be able to start typing things out (sure, it won't be very good, but editors aren't good enough to solve that problem yet).
In a corporate environment, you want that. I should be able to sit down at your workstation, or you at mine, and we should have a common language that we can speak. With a standard editor like Sublime, you can say, "okay, now open your sidebar, and go to this directory and open that file, then scroll down until you're at this function."
There is some power to having that commonality. You don't have to flame me for using emacs, and I don't have to deride you for using "modal editing, whatever that is," every time we get together to edit code. And if I sit down that your keyboard there's a very good chance that we will have the same keybindings, even if it's my very first day as a brand new programmer fresh out of college/bootcamp at your company.