I was going to work out the annual costs of using a dishwasher versus washing dishes in the sink but someone has done a pretty comprehensive job already: here. (From 2012 but the prices shouldn’t be too different from today.)
In summary: she says there’s not a huge difference in energy costs if you hand-wash the dishes in running water versus using a dishwasher (~£40 / year).
However, I worked out that on our current tariff, 2.5 KWh to heat running water every day would cost ~£125 / year. (Using a basin would cut this hugely.)
There’s the cost of dishwasher tablets (About 20p each), which adds up to about £70 a year — more than the cost of the electricity the machine uses.
Then there’s the price of the washer in the first place. Importantly, a lot of emissions will have been created in manufacturing it.
I’ll have to look further into this and see if I can come up with more definitive information.