I would be using a dremel and a cutting blade and run around inside the button housing and cut a ring right round. Off comes the cistern lid then just replace the flush button for a suitable new type or worst case replace the flush valve and button.
Stuff that for a game of soldiers, if it's being replaced then break out the old button, it's only plastic. Bit of wood and a plain screwdriver and snap the edge off all the way around, it'll become apparent how it's fixed once the lid's off.
If you need to find a tradesperson to get your job done, please try our local search below,
or if you are doing it yourself you can find suppliers local to you.
Select the supplier or trade you require, enter your location to begin your search.
Please select a service and enter a location to continue...
Are you a trade or supplier? You can create your listing free at DIYnot Local