Best I remember I locked it with the key that works to lock/unlock the glass cabinet (don't remember having another key). The key will turn all the way to the right but only partially to the left. I talked to a person who works with old furniture who suggested tilting the piece to the side and also front and back and see if it would loosen the tumblers but no luck. Wondering if anyone here has experience with/suggestions about this. Here is a picture: https://www.ebth.com/items/3747688-writing-desk-with-glass-cabinet
Bit-key locks of this sort often have a fairly simple mechanisms. In the very simplest, there is simply a linkage to or notch in the lock bolt which the bit (the part at the top of the key) pushes one way. There may be obstructions ("wards") which require specific areas of the key to be cut away so it can rotate fully, which is why there is a notch in the bit ... but basic locks may have shipped with a notched key despite having no wards at all.
Those simple locks were easy to pick, or to open with a "skeleton key", so fancier mechanisms were developed. It is possible you have a simple lever-mechanism lock, where part of the bit lifts a spring-loaded piece to release the bolt while other parts of the bit engage with the bolt to extend or retract it. I wouldn't expect to see anything more complicated than that in most furniture, especially given the simplicity of that key (assuming it's the right key).
In most locks of this sort, the key makes a complete turn one way to extend the bolt and a complete turn the other way To retract it again. If the key is not making a complete turn in either direction I would suspect that something is stuck.
I think the first thing Is suggest an amateur try is to turn the key as far as it goes one way, continue to apply gentle turning pressure, and rap the front of the cabinet at and around the lock. If you're just dealing with stickiness or dust, that may shake things free. Probably the direction to try is the one in which the key rotates less, but alternating may also help free things up.
If that doesn't do it... You may indeed have the wrong key and be running into wards or levers. Or it may just be stuck worse. Or key and/or lock may be worn, or there may be random dirt in the lock. (Some kids try to feed machinery...)
(This touch keyboard is going to drive me mad yet... OK, I hear you all, "too late!")
((Late thought: Sometimes the problem is that stuff inside the desk has shifted and is pressing against the back of the door, binding the lock bolt. If so, thumping it as above may help, but so might tipping the whole cabinet back. You might have to dismount the upper cabinet to do this safely, though; often the two are designed to come apart for moving and if so that isn't really a structural joint.))