This has been discussed many times around here in terms of using a US outlet strip overseas. It's generally OK as long as the outlet strip doesn't have other electronics (perhaps a surge suppressor) that can be fried.

Using an outlet strip designed for 240V in a 120V environment might be more problematic, however. Higher voltage wiring requires thinner wiring to carry the equivalent wattage than it would at a lower voltage. You need to ensure you're aware of the amperage rating for your outlet strip and don't exceed it, remembering that the power available will be half of what it would be in the UK.

That said, let's work the math (or do you say maths). I believe the UK standard is 13 A at 230 V, or a whopping 2990 W - almost double what the standard US wall socket can provide. Plugging in your outlet strip in the US will let you do 13 A at 120 V, or 1560 W. This is equivalent to a standard space heater or high power hair dryer. You won't come close to tripping the circuit breaker by plugging in various electronics.

So to recap - you'll be fine.