I had the opposite experience - I had an old washing machine that kept consuming more and more, so I decided to replace it a few years ago. The difference in my electricity bill was noticeable, but I'll warn you it really depends on how many times a week you wash and what cycles you use. If you wash a lot, the savings show up pretty quickly; if you wash little, it'll take longer to break even on the investment.
What nobody tells you is that besides the inverter motor, pay attention to the energy rating. Having an inverter is one thing, but having an A-class or higher is another. And here's the trick: manufacturers sometimes sell inverter models that are still expensive to run because they have huge capacities or unnecessary features. Compare the annual consumption in kilowatts on the label, don't just rely on it saying "inverter."
Another practical thing: before you change your washing machine, check if your electrical installation is optimized. It sounds weird but sometimes an old machine uses more because there are problems in the installation that a new one detects better. That said, if your washing machine works fine but visibly consumes a lot, changing it to a decent new one if you can afford it, is worth it. You'll probably see the savings in one or two years depending on your habits.