Let's be difficult: it's not exactly the 'next day' you will receive your precious gems, it is
the next shipping you will receive them. What this mans, and now comes the fun part, is even when you send your gems away at 8 am in the morning, you will receive them back one hour later, at 9 am; the next shipping.
If you have World Adventures your gems cut in a foreign country will be taken with your sims the moment they arive at home. So this means that if you send 10.000 gems away to be cut right
before you go back home you will still have them cut in your sims inventory when you arive home: all 10.000 of them.