Can I use a cheeseboard as a cutting board? Nowadays I see that most cutting boards sold are made up of several pieces of wood that are stuck together with wood glue. I'm not so sure of the safety of the wood glue that is used as most of the ones sold in my country are manufactured in China and not subject to FDA approval: Safety of glues in wooden chopping boards

I noticed that most cheeseboards sold are however made from one piece of solid wood. 

Would I be able to purchase a cheeseboard and convert it to a cutting board instead?

This blog post shows people how to create cutting boards from pieces of wood that can be purchased from a hardware store and glue it together to create the final product. The cutting board is then rubbed several layers of mineral oil, walnut oil or beeswax to protect it and allowed to dry overnight.

Would it not be the same to purchase a cheeseboard that is already cut to an appropriate size and then just rub it with several layers of mineral oil, walnut oil or beeswax to protect it? Are there any disadvantages to doing taking a protected cheeseboard and using it as a cutting board?
You certainly can use a "cheese board" or other single-slab piece of suitable wood as a chopping board. 

What you might want to consider is that there is a reason why even expensive, good-quality chopping boards are made from multiple pieces instead of one big slab:
Wood can warp and / or develop cracks if exposed to changes in humidity or when drying out. Using multiple pieces glued together in alternating directions counters that effect to some degree. This does not mean that your one-piece idea won't work, but that you should take good care of the board and avoid long exposure to water (don't soak it) or dry heat. Keep it well-oiled, clean and towel dry it immediately after use and leave it in a well-ventilated area between uses.