If there are electronic circuits for controlling the timers, feeding pumps, etc, then they can all be for 110V only - the lights would work but you'd risk destroying all of the other circuitry. Do you mean to say that you used the US system with UK power and *as a result* blew the bulbs, or do you mean that you've been using it all along with a step-down transformer and your bulbs have just naturally gone at the end of their life?
Check the aerogarden for the stamp/sticker that shows the power requirements - lots of appliances these days accept dual voltage. If it does it will say something like "Input 110-230V 50/60 Hz" (rather than 110-120V 60Hz only). If that's the case then you can use it on UK power with UK bulbs or on US power with US bulbs - no problem. If it's US power only (110-120), then you need to get those bulbs, or bulbs which are rated for dual-voltage, and continue to use the system on the step-down converter.