You have an impossible task. Rejuvenate your CV, and find your next job.
Seriously though, start with a budget. Until you can secure funds you cannot do anything and the budget will tend to direct what you can accomplish next. Once you have cash, find the oldest piece of hardware in operation and start with that one. You will have more failures based on hardware than you will based on unpatched OS's. Disks are your primary concern in this realm.
Second, after you've completed a few of the more horrendous back-end server migrations, the desktops are next. This is a political move. It will endear you to the user community and this will make additional funding possible. If you focus entirely on the back end, you will run out of support and therefore money long before you can complete the task. You may have to do this step by department, so make sure that your most supportive users get their upgrades first. As I said, this step is entirely political in nature. You will not be able to perform all the upgrades in this step, so be picky.
Third, address the network. Given the health of the server architecture you've described, I suspect that even gigabit-Ethernet is foreign to your environment. Make sure you can build in redundancy along the lines of 802.3ad (LACP) etherchannel connections for all things. Redundancy is your top priority in a network refresh. Basically there are two (2) of every component, each of which is connected to two (2) others.
Fourth, take the remaining servers in order of business impact, most first. This will give you the opportunity to introduce the user community to the concept of "maintenance windows". It will also allow you to engage top management in the upgrade process, which should allow you to re-negotiate the budget; which will be woefully inadequate at first.
Assuming you've made it this far (doubtful) go back and finish the user PC upgrades.
Then prepare to do this entire process again in about three (3) years. Perhaps five (5) if you are lucky enough to get the funds needed to buy things which have significant life. Leasing is also a good thing here because it forces the refresh once the lease terms are fulfilled.