Cat suddenly won't come out of hiding

My cat is old, he's 14. We recently got 2 puppies and they've been with us for almost 2 months now. My cat was stressed at first, but he seemed to calm down quite a lot after a month and he seemed fine. He tends to prefer being upstairs to downstairs most of the day, and to keep the puppies out of the rooms we close all the doors upstairs. My cat likes to sleep in one of the bathroom closets, and it wouldn't be the first time we' d accidentally locked him in there. Yesterday it took me all day to realize I hadn't seen him. He must have been trapped in the bathroom for at least 10 hours, I hadn't noticed until it was food time and he didn't come running. So I go to let him out and he's very reluctant and meowing in a rather distressed way. He finally runs out as if he's scared to death, and goes right under the bed. I try talking to him, but he just meows loudly and stress-sounding at me in response. I assume something is wrong, but I don't know what. I thought maybe he was just traumatized from being accidentally stuck in the bathroom closet all day, so I brought him food and he came out to eat and drink, but that was it. After that it was right back under the bed. My cat never goes under the bed. Not unless he's scared., which is very rarely. I've never seen him refuse to come out when everything is safe and sound, though. I decide to see what happens in the morning and maybe he just needs some relax time. Next morning nothing has changed. He did more destinations from under the bed to behind the tv stand. Both areas no one can physically reach him in. But he will not come out for anything other than food and water, and you have to put the bowl as close as possible or he won't budge. I notice he hasn't had a bowl movement in over 24 hours, and he peed on my bed last night. I moved his cat box up here so hopefully that won't happen again. I've also noticed he's drinking a lot of water. He's typically rather obsessed with water as is, but he seemed much more interested in drinking for a long period of time than eating, but he will eat a sufficient amount. 

I'm wondering if it's Kidney Disease, but none of it talks about your cat just hiding away from the world when 2 seconds ago he was his usual self, lively and social. His sudden seclusion is really puzzling me and I'm basically preparing for the worst. 

If any of this sounds familiar to you, I'd be grateful for replies. I just want to understand what's happened to him for a bit of clarity before I can take him to see a vet.

Cats are very good at internalizing stress, which can then manifest as health problems. I would interpret any change in behavior as a good reason to schedule a vet appointment. 

As you probably already know, cats are extremely territorial.  Any new addition to the household can be traumatizing, though it doesn't have to be. The very best thing you can do is make sure there is plenty of vertical space for him to claim in living areas. Roomy carpeted shelves for hanging out on, smaller shelves to serve as "stepping stones" to high places of sanctuary, cat trees, etc. You can get really creative with stands and bookshelves. Hauspanther is a good place for inspiration 

Think of it as creating a cat "superhighway" around living spaces. Ideally he should be able to navigate the room without needing to touch the floor. It's all about giving him back that lost sense of security. If he knows he can hang out with the humans without having to dodge puppies, he'll be much happier (and consequently healthier!).