Do you know how much your bill would be for the new house without a meter? In our old house we paid £35 a month on a metre (me, oh and 21 month dd) similar to you in that washing machine was once a day and mainly showered. New house is £43 per month un metered, we're ttc soon so aren't going to ask for a metre as it'd probably end up costing us more
firstly, see if you have a choice; some authorities (not all) put in a meter when a house is sold.
a full dishwasher uses less water and energy than washing up. A power shower of more than 10 minutes (which is way too long anyway) uses more water than a bath.
if you do have a choice - in your situation you should save money if you are in the south of the country where 'water rates' are much higher. Doesn't work for all areas. You may also be able to try a meter for a year and go back to un-metred if the bills come out too high.
We are in SW so most expensive water rates in the country. The are 6 of us. We are not especially careful with water but don't water the garden/wash the car etc. We are on a meter.
My mum was asking what we pay as she is thinking of going onto a meter. We currently pay similar to her Yorkshire water non metered annual rate! She lives alone and also does not water the garden/wash the car.