Empty lotion tubes and bottles (without the pump) can be recycled. Make sure they are empty so you don’t contaminate the recycling stream. If you can’t get all the lotion out and/or are worried cutting the tube open may cause the pieces to jam the sorting machines, then put the tube in the trash.
Check your local provider to see what they’re accepting.
It’s not recommended to put the pump or spray mechanism of any bottle in the single stream recycling because that is a composite/mixed item of plastic and metal, and those little plastic bits are actually very small and can jam the sorting machines.
Lotion can be hard to find a best practice on because it usually comes in a bottle with a pump or in a tube. Both have issues and are more difficult to recycle. If you can find a tub or a lotion bar those will be easier to recycle and compost, respectively. Many people also find oils like coconut or olive to work well on their skin. I tried coconut oil for several years and it never worked completely as well as a lotion does. I was also always worried about it absorbing enough to not stain the couch or my clothes.