This is a wiki answer, so please amend with additional solutions, or improve the ones below:
CPC Tape Software
Tape images are normally downloaded as a .cdt file, the same file format as the Spectrum .tzx tape format. Though these do not contain any sound data, they can be played back on an Android or iOS phone, using an app such as 'PlayZX' or 'Speccy Tape'....
For the ZX Spectrum:
Sorted by amount of non standard modern equipment needed (less to more):
A smartphone with SpeccyTape (IOS, unfortunately, now unmaintained and not available for 64 bit devices) or PlayZX (Android) and a stereo to mono cable.
TZXDuino / TAPDuino: hobbyst grade gadget (built and sold in several places) that essentially does the same as ...
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
Looks like it uses WD1770 I am not familiar with it but first check if WD1770 is compatible with IBM format. If yes then you can read the discs directly on PC ... If not you need HW capable of reading the disc.
So in case of PC compatible you just image it using direct sector access in C++ (or any other language or utility) instead of file format. After you ...
I haven’t done this but I believe you could instigate a DMA transfer via the 2MHz ‘tube’ interface and write the image directly into RAM. That would require additional hardware but no modifications to the machine itself.