Use the old as a reference point, and replace with a lampshade that closely matches both the bottom diameter and height.
Use the existing harp as a reference point, and look for a lampshade with equal or slightly greater height. Then use the information in the tips below to confirm that the diameter will be a good fit.
If you’ve found a fabulous pair of antique vases, with no harp, or a harp and old lampshades that are oddly sized, your best bet is to start fresh, and refer to the tips below.
First measure the base, from the tabletop to where the neck meets the base of the socket. Your harp should be somewhere between 3/4 and 4/5 the height of the lamp; it should rarely be the same, and certainly never larger.
You might also measure from tabletop to the base of the socket, and multiply by 1.25 to confirm that the diameter will result in a pleasing proportion.
Assuming that they are not meant to have very small shades perched on top (like some buffet lamps), just about anything smaller in diameter than the height of the base will work, and look best when the stick is about twice the height of the shade. It is unlikely that you would ever find yourself needing a lampshade larger than 16″ in diameter for a tabletop stick lamp, unless you’re going for a flamingo effect. Empire shapes seem to work best, as the taper at the top balances the bottom.
A good rule of thumb is that the smaller the bottom diameter, the more flared they should be, like a coolie hat. As the bottom diameter increases, less tapered and more drum-like frames can work. I have made both: diminutive shades that are like little hats, and large and jumbo shades, up to 24″ in diameter.
Chandelier lampshades are all more or less similarly sized: they have a torpedo clip fitting, and are typically about 4″ high, and under 5″ in diameter, too. And because they are often more diminutive in scale, or at least in wall projection, vintage wall lights are often better paired with chandelier shades.
Larger sconces or wall lights are a bit trickier as the key measurement is the projection from the wall, and if they are double or triple light sconces you must also take into account the distance between the lights. Remember to use half the diameter measurement when determining whether a shade will fit: a sconce with a 6″ wall projection to the light will easily take a 7″ diameter lampshade, as the shade will extend only 3.5″ towards the wall, leaving a good bit of space.
Finally, swing arm lamps of the sort you’d find above a sofa or reading chair often take a 12″ diameter small empire lampshade with washer fitting at top.