How can I polish slate?

Here's one article that might help you - https://www.doityourself.com/stry/how-to-make-cheap-slate-tiles-shine

Slate doest get glossy, not it's nature, too potus. You can use the wipe on tile sealer that does Enhansing to add a slight sheen. The oils you asked about may just look like a grease stain. If there is a spot not so noticible, give it a try. My concern is some oils just get greasy and sticky.

Hello, I recently used both the stone cleaner and sealer my slate kitchen ( lazy susan and coasters) items and fireplace hearth using these products. The results were wonderful, nice luster and depth, sealed but not a wet look high gloss finish.

It was the exact finish I hoped for to have protection while cleaning the firebox -I was looking for a seal moreso than a polished glossy finish. If you’re looking for a high-gloss finish this might not be the perfect option for you. Perhaps a dealer like Floor and Decor might have a high gloss option better suited for your personal aesthetic.

Tung oil it is less drying than the Walnut & not as dark; you can also use Teak oil(also less drying) or you can wax it much longer process need buffer TUNG OIL https://www.youtube.com/watch?v=kZjJYYOC8uM they do make Slate oil but not readily available in large quantities https://soapstoneproducts.com/product/slate-oil you'll need to do it in layers applying a little at a time lightly until you get to desired gloss you want if you apply to much all at once Slate won't be able to absorb it as needed then it will get "sticky" make sure to use lint free cloth(s) I bet it's beautiful.