I couldn't find technical information or an existing emulator but via your links I found this high-resolution image of the interface's board.
From there I notice the following things:
- the disk controller is a WD1770, that's the big one on the left;
- the two large chips on the right both half underneath the edge connector are an 8kb RAM and an EPROM; and
- everything else that's visible is 74-series logic.
It's a safe bet based on the architecture of the Spectrum that the interrupt button switches in the on-board ROM and triggers an NMI.
The ROM images linked are 32kb and 16kb in size but the former just seems to be 16kb data copied twice you can also adduce that the EPROM is 16kb, in which case there might be some guesswork in figuring out when and where that 8kb of RAM appears.
I found old mentions in comp.sys.sinclair that the Microdrive emulation added to a later version was partly hardware based, but seems to have come with a board revision. So an emulator author would probably start with the older EPROM in the hope that the extra hardware wasn't yet present.
Otherwise, the good news is that the disk controller is an extremely standard part that just does vanilla IBM encoding, and based on the advertised floppy capacity of "thirteen 48kb games per disk", you can assume the physical format of the data on disk is MFM (i.e. regular PC 'double density').
So: you might be able to image your disk using a plain USB disk drive (unless you find one that supports high density floppies only) as MFM is essentially the only thing they support — but they're often restricted to 512-byte sectors with fixed numbering so this isn't guaranteed. Might be worth a try before looking around for a less-common solution though. This MSX-related page gives a guide for Linux and macOS though the Windows section is empty. Some slight more research might be necessary there.
Failing that, look into solutions that can image floppies for a machine like the Amstrad CPC, as those should handle different sector sizes and addressing, and the common CPC-related file formats can retain all that information. If you have an old enough PC to have a pre-USB floppy-disk controller and drive and an OS like DOS that gives software unfettered access to the controller then that should do the trick. Or low-level disk controllers now exist for exactly this problem, such as the Greaseweazle mentioned by scruss, the Kryoflux and others.
Starting from a working Spectrum emulator it then shouldn't be too much of a trek to get the Swift disc interface added and to read your floppy.
What software should write to a WD1770 is very well-known so you could isolate those addresses fairly easily, and pretty much the first thing the NMI routine will need to do is backup the current display so you'd likely be able to figure out the RAM question without too much dilemma. At that point it might well be as easy as just plugging in your disk image.
A lot of the time when these slightly more obscure interfaces don't get emulated it's just because no test data is available; it's very possible that you'd be able to get an existing author interested just by having a disk image. Otherwise a Spectrum emulator is easy to knock up so someone like me who has the generally correct sort of code kicking about should be able to help.
Alternatively, since my search wasn't especially thorough, you might find that the thing already is emulated — in which case please let me know so that I can change 90% of this answer.