I'm afraid you may have a real problem here, Chris. I do not have a functioning 5.25" floppy drive. I have a 3.5" drive, and that's a rare bird. I'm down to one readable disk for that.
You've got an actual 8086? With an actual 5.25" drive? Functioning? Running DOS, I presume? What version? Have you got interlnk and intersvr? If you can come up with a "nul modem" cable, that might be a plausible way to transfer files to your valuable antique.
Recent versions of Nasm, nominally for "dos" are in fact built for 32-bit "extended" dos. Won't run on an 8086. The last version of Nasm that ran on a 16-bit machine would be 98.? I don't recall the exact version. The last time I tried to run Nasm on a 16-bit machine - an 8088, actually - it crashed. I think it may actually have required an 80286. Not sure on that.
Up until fairly recently, we carried some really old versions of Nasm here, but when I just looked we don't seem to go back beyond 0.99. Dunno what happened there. So you may have a problem finding a build that will run on an 8086 even if you can find a way to transfer it. Are you in a position to compile Nasm yourself on your oldie, if we can come up with suitable source and a way to transfer it? I may have some pretty old versions on my machine - no guarantee.
I would suggest that you "cross assemble" on your modern machine, and transfer to your 8086 either with interlnk/intersvr or find a 5.25" drive for your modern machine. Nasm's output should run on the old machine no problem - if you do it right.
If you want to assemble on your old machine and can't get Nasm to run on it. Eric Isaacson's A86 might work, or an assembler called "arrow" - or maybe you can find versions of Masm/Tasm old enough to work. Good luck with it!
Best,
Frank