In general, hand sanitiser is safe to use around your jewellery, EXCEPT when it comes to pearls and opals. You may find that sanitiser leaves a film residue on the surface of your jewellery, in which case you can follow the steps above to clean it off (if you’re out and about, a quick polish on your denim will do the trick).
Be careful with bleach and household cleaning products – ideally, take your jewellery off before cleaning, or wear gloves. Pearls and opals are, unsurprisingly, the worst affected by cleaning products.