Put quite simply, they will just have to and that means that as their parent, you will as well.
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You can visit a holistic dentist, who will take into account the child's emotional wellbeing. You may also decide that it is in your best interests, and in the best interests of your child, if you talk them through how things will be slightly different whilst you are away from home. You can explain that you will eat in different restaurants, sleep in a different bed, and you may even see a different dentist there too. Make sure that you do not impress this as a likely event; just tell them that it could happen.
Your next port of call is to have a list of three, if not five, child-friendly dentists that are going to be in the area where you are going to travel. Having their direct phone numbers on hand at all time will definitely alleviate some of your worries. That way, you know that should the very worst happen, you at least have the contact details of people that will be able to help you and child at once.