I wish people would be more specific in terminology. There are at least three parts of a lathe that could be called "gearbox:" headstock transmission, quick change transmission, and apron transmission. All have their separate ways and means of service for water extraction. On some lathes there may be more - like a two speed handwheel on the quill, the index reversal, and sometime an under-drive headstock transmission.
The head stock opens quite simply by removing the top. Inside is the spoindle transmission, shifters, etc all there to see including the sump which may be ribbed with little ponds of water in each depression which have to be idividually sucked clean. The quick change may have to be removed to be cleared of water. I don't know which Monarch lathe you are wrestling with and I'm not that familiar with Monarch anatomy to begin with. However they are well designed machine tools, built for ready maintenence. Reference to the manuals should indicate a path not difficult for anyone with mechanical experience to follow. The apron may be inspected by positioning it about mid-travel along the bed, supporting it by blocking it from underneath, removing the attachment bolts to the carriage and lowering the apron on the elasticity of the lead screws and feed shafts to clear the crossfeed gear where it projects above the apron joint. Move the carriage sideways down the bed and you can inspect the apron internals and clean them of debris, coolant, and water.
What's de-watering oil? I never heard of it but that doesn't mean it doesn't exist. Sounds like a Godsend to the struggling mechanic.
Don't refer to procedures like "flush with diesel" lightly. Think it through. If a transmission is compartmented so that multiple lube changes still brings off water in the drained oil, diesel will only complicate the problem. Diesel has very limited lubricity. Even small amounts of diesel contamination inhibits the film forming properties of the intended lube oil and greatly inhibits the boundary effect in lubrication. It takes many oil changes to dilute the diesel remainder sufficiently to safely allow full transmitted power. A thoughtless diesel flush could do more harm than good.
Water in complex equipment items can be "baked out" but to do so is a hassle if the item of equipment cannot be fitted in an available oven. You can extemporise an oven with 2" foam sheets or a light frame and glass batt insulation (provided has a vapor barrier to prevent air movement thruough the thickness.) Surround the item with isulation, taping the joints and leaving air space between the insulation and the item. Provide a 2" hole top and at the bottom edge for limited air circulation and a board to cover. Install a circulating air heater and a temperature sensor - preferably a temperature controller's thermocouple to be attached to a solid mass on the item. Ramp the temperature up to 220 degrees and hold for a couple of days. This will not hurt the electrics, rubber,. or plastics but it will smell. Water will be driven off as vapor.
A salvaged electric furnace element set and blower works very well as a heater and air mover and its relays can work from the controller's NO contacts. This is a heroic measure that takes time but it works very well. It's long been used in electrical equipment. I've used it to de-water a transmission on a diesel truck, dry out many a drenched electrical panel and M/G set, and a few flooded machine tools equipped with complex hydraulics.
Water intrained in oil is not a loss. Most mineral oil fomulations will drop emulsified water. Let your drained oil stand for a day then pour it off carefully into a clean container. A chamois soaked in oil and placed in a funnel over a liner of expanded metal will exclude water from oil filtered through it. Oil passes chamois slowly. Heat the oil to 120F and the process goes much quicker. Out in the boondocks, aviation gas has been strained through chamois funnel for a hundred years.
Once the oil has been separated heat it to 250 F (in batches if you have to) and the water will evaporate off. The usual precautions apply. One thing: DON'T heat oil with visible water in it. It may explosively pop hot oil.
If a machine tool transmission is suspected of water contamination you better do something positive quick. Rust waits for no-one.
If it was simple anyone can do it.