Is there any way to get Bluetooth re-enabled without a USB keyboard/mouse?
If you have accidentally turned off Bluetooth thereby disconnecting Apple Magic Keyboard and Magic Mouse/Magic Trackpad, simply hard reboot your Mac by pressing and holding the power button.
Then restart your Mac, and it will trigger the wizard for connecting keyboard and mouse which will in turn, enable Bluetooth.
Do note that hard rebooting may not be the desirable approach as it may cause you to lose data.
Could I remote in from my iPhone somehow?
If you already have screen sharing configured, it's possible to use any of the various VNC apps available on the App Store to remote control your iMac and thereby easily turn on Bluetooth.
If I have to get them, will a wired keyboard be sufficient to enable Bluetooth without a mouse?
Yes. Open System Preferences, and use the keyboard shortcut Command + F to highlight the search field. Now enter keyboard shortcuts and press Enter. You'll be presented with keyboard shortcuts preference pane. Now use the keyboard shortcut Control + F7 (You may need to use the fn key), to select All controls setting.
Now switch to search field again using the keyboard shortcut Command + F and type Bluetooth. Use the Tab key to move focus to Turn Bluetooth On button and press Enter or Spacebar to toggle.