I have an old wooden desk. The drawers are locked and the top drawer somehow controls all. The top one opens fine, but it prevents the 7 others from opening.
This desk is designed to be able to open the top drawer without a key but you will need a key for the top drawer before you will be able to open the other drawers.
Boy this post makes me miss my dad who even though he was not a locksmith, he could open any locked item in a short time. Most locks, especially the old ones are pretty simple, despite the complex multi locking drawer thing. He used to call the set up you are describing as an illusion of safety as once you pick the lock, you are in the drawers toot sweet!
This post is very rambling, but somewhere in the middle of it. They talk about safe and simple ways to open then suggest taking the lock mechanism to an antique store to match the key.
My dad would just take the lock innards to the flea and look thru junk boxes for rusty keys and almost always found a match for pennies. Apparently with skeleton keys (if that is what you have) there were only a few main cuts. If he found an almost, he would saw off a cut and make his own!
So I guess long story, long, if you are careful and even remotely handy, you can YouTube video to pick it with very basic tools and then find a key at the flea for cheap! It will be an adventure!
I have an old student desk that has a lock triggered by the center drawer. If you pull the center drawer open slightly it triggers the bar to release the side drawers from the locked position.
Call a locksmith. You dont want to devalue this furniture by damaging it.