It sounds like your son is going through two things. First, a little seperation anxiety, normal for this age. Give him extra cuddle time, but also make sure he knows it's okay to have independent time, stay where he can see you, keep talking to him, even while you can't be holding him. You want him to get more independent at this stage, not less independent. But if he needs to be held, by all means do so.
Second, he may be getting ready to go to one nap. This will mean a change in routine (just when you thought you had a good one!) It will take awhile, maybe a couple of weeks. You want to drop the morning nap, not the afternoon one. Ideally, he will eventually take a 2-3 hour nap after lunch time, you will find out how blissful this can be! You will need to keep pushing his morning nap back, a little at a time, and get to a point where he will eat lunch, then sleep. This may mean eating lunch at 11 am, but as he gets older, he will be able to stay awake longer in the morning.
There's no reason for a one year old to give up napping, they need that rest time. So if this is important to you, stick with it. It took me lots of time and perserverance to get my son into a good routine, he's 2 and 1/2, and still naps every day. He loves his nap, and so do I!