Make up some drawings for your ideal unit, make a cut list and send it to cutwrights. They can prime the panels for you.
Edit - you will need to source the sink yourself.
If I do this I might as well just box the pipes in and reattach the old wall sink - I’m hoping for a minimal effort job!
I appreciate that but it sounds like you're after something quite bespoke. Therefore there will always be a cost or effort price to pay. There are plenty of built in vanity units that are ready to go but I've never seen one that is ready to paint.
@Dramatic_Hammer started
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Make up some drawings for your ideal unit, make a cut list and send it to cutwrights. They can prime the panels for you.
Edit - you will need to source the sink yourself.