Not good news I'm afraid. Platinum is the most expensive it's ever been. I had mine done recently and it worked out about £100 per ring. However, if you're only going up a size or two, then they might be able to stretch the wedding band, provided it is plain (without stones) and hasn't been stretched or resized before.
In 2003, the price of platinum was between $600 & $800 per ounce. This year it has been between $1500 and $2300 per ounce. Quite a remarkable price rise in 5 years and explains why getting a tiny piece put into a band is so expensive, compared to what you will have paid for the ring when you got married. If you got married more than 5 years ago, then platinum was even cheaper again.
If you only want to go up 1 or maybe 2 sizes, then they usually 'ease' or stretch the metal rather than using any extra, iyswim. And in this case it really shouldn't cost more than £15-£20 per ring. If it does, you are being ripped off. Shop around & use an independent jeweller.
They're not supposed to stretch rings with stones though, as the integrity of the setting can be compromised. OP was asking about engagement rings as well.
No, you can't usually get paid for metal taken from jewellery. Something to do with unhallmarked metal. Also with platinum, it's very difficult to rework, so is rarely reused. I had rings made smaller before and whilst they gave me the bit they took out, it can't be used again to make it bigger because platinum is too hard to rework.