Warning!
When it comes to OLD, original .38 Long Colt revolvers --- BEWARE.
Many of these fine, old guns were chambered with no shoulder in the chamber. This allows them to chamber .38 Special and .357 Magnum.
Just because it chambers doesn't necessarily mean it's safe.
However, in the case of low-powered (and low-pressured) .38 wadcutter loads with black powder you could probably get by provided the revolver is tight.
But why take the gamble?
As to the .38 Long Colt in a .36 Navy conversion.
The original .38 Long Colt cartridge used a heeled bullet. That is, a bullet with a stepped-in shank on the base. Only this part was loaded into the case. The rest of the bullet was outside the case and larger than this shank.
In the case of the .38 Long Colt, bullet diameter ranged from .376 to .381 diameter, according to vintage samples measured in "U.S. Cartridges and Their Handguns" by Charles Suydam.
I'm not sure what the bore diameter was in the older guns but it was probably slightly smaller than .375 inch.
The inside diameter of the .38 Long Colt case is about .355 inch. It was this very case that was lengthened slightly to create a new pistol cartridge about 1902: the .38 Special.
In the .38 Special, a bullet of about .357 inch is used.
For best accuracy in a .38 Long Colt revolver, use a heeled bullet. Moulds to cast your own heeled bullet are available from some specialty mould makers, such as NEI and Rapine.
If you use a heeled bullet, the case must be trimmed back to accommodate for the longer length of the bullet.
Heeled bullets are assembled onto the case without lubricant. Then, when the cartridge is assembled, the entire cartridge is up-ended up and the bullet is dipped into melted black powder lubricant. The bullet is held in this melted lubricant for a moment, then the entire cartridge is withdrawn and stood upright on a piece of wax paper.
This is done one-at-a-time for each cartridge.
Now, you can see how time consuming it can be to assemble ammunition with heeled bullets. I know, I make heeled bullet ammo in .32 Long Colt caliber for use in a Marlin Model 1892 rifle.
Perhaps a soft lead .38 Special bullet with a deep hollow base would bump up, upon firing, to fill the larger bore. I know that it works poorly in the .32 Long Colt and that heeled bullets give best accuracy.
Here's a thought:
Try experimenting with a conical bullet intended for the .36 Navy. Most such bullets already have a heel for an easy start in the chamber.
Two designs that come to mind are the Buffalo Bullet and the Lee. The heel would have to be about .358 diameter to grip the case securely.
Whatever bullet you use, it must be of soft lead when used with black powder. And, its lubricant must be made for black powder. Lubricants that may be used include Bore Butter, SPG, Lyman Black Powder Gold or a variety of home-brew recipes.
My favorite home-brew lubricant recipe is 1 part canning paraffin, 1 part mutton tallow and 1/2 part beeswax. All measurements are by weight. I use a kitchen scale to measure 200/200/100 grams of ingredients and melt them in a quart Mason jar placed in 3 or 4 inches of boiling water in a pan.
This makes an excellent black powder lubricant for a variety of uses: bullets, felt wads, patches, etc.
Frankly, using a converted Colt Navy requires some effort to keep it well-fed. You just don't pop down to the store and buy a box of black powder cartridges, assembled with heel bullets, soft lead and black powder lubricant. It's requires a commitment.
But if you can get Navy converted to fire standard .38 Long Colt ammo (without a heeled bullet) that uses .38 Special bullets of .357 inch or so, that may be the way to go.