Leave the charger at home.
Your charger will not work in the USA, even with a plug-adapter that allows you to plug a BS4573 plug into a normal US "NEMA 1-15" outlet.
How to tell
The base of the charger says
220-240 V AC ~ 50-60 Hz 0.9W
UK vs USA
You are using this in the UK on a nominally 230 V AC supply at 50 Hz
Normal† outlets in the USA are nominally 120 V AC at 60 Hz
So you are OK on frequency but the US voltage is way outside the range for which this charger was designed. If the charger could be relied on to do anything useful at 120 V AC it would say so on the base. It is unlikely to even charge the toothbrush at a slower rate. Most likely, it will do nothing at all‡.
It is unlikely the charger would be harmed or catch fire. However any charger you buy in the USA is likely to be dangerous to use in the UK unless the markings on the charger explicitly say it is usable with up to at least 240 V AC.
As others have said, you can do any of the following:
- Buy, in the UK, a toothbrush with a charger that accepts 100-240 V AC
- Buy, in the US, a toothbrush with a charger that accepts 120 V AC
- Take a manual toothbrush (probably a good idea as a backup)
† US homes also often have a 240 V AC split-phase supply for high-power appliances such as laundry machines. However these use a completely different type of outlet and plug. You will most likely not have access to these outlets and will not be able to buy an adapter that will allow you to connect a BS4573 plug to any of the multitude of different types of 240 V outlets.
‡ The reasons I expect a 240 V charger to do nothing useful on a 120 V outlet are as follows:
The charger and toothbrush-base are two halves of a transformer. The charging relies on inductive coupling between coils in each part. The effect is to reduce the mains voltage by the ratio of the number of turns in the two coils. The toothbrush then probably has a voltage regulator IC (and some passive components) that expect to receive say 7 V AC and turn it into say 5 V DC - or whatever is needed to charge the rechargeable battery in the toothbrush.
A ratio of coils that turn 240 V AC into 7 V AC will turn 120 V AC into 3.5 V AC inside the toothbrush. It is likely that this will be too low for the regulator to produce 5 V DC from. You would need a more expensive boost-regulator for that and manufacturers generally don't waste money that way.