Is the UPS a battery powered UPS, or is it just a power strip? If the UPS is battery powered, you should be fine, the neutral (common) buss is grounded at the source (house panel, somewhere) so the UPS will ground as it needs to. If it's just a power strip, you'll maybe run into issues. Ground wires are designed to be non-current carrying conductors, they are there for 'extra' stuff accumulated in the house. Static electricity, radio frequencies etc, basically a bunch of electrical noise that's best gotten rid of back to the panel. The other 2 conductors (240v is 2 'hots', 120v is 1 hot/common) are both current carrying, depending on what's in the line you can get 'shocked' from either. An ungrounded system leaves no room for accidents. Bad voltage, spikes, static all have no room to leave, no ground. Touching the frame, you make the ground, so you become the path, you get shocked, the frame gets electrically charged and anything attached to the case gets spiked in reverse, the electrical discharge 'goes in the back door' so to speak. This is bad. It's a worst case scenario, but generally speaking, an ungrounded piece of electronic hardware, be it a TV, radio, pc etc is just an expensive accident waiting to happen.
If you don't have a battery powered UPS, I strongly recommend that you get one, sooner rather than later.