You might consider the Pi 4 Compute Module, it would be easy to integrate onto another PCB. Otherwise a 4B will work just as well and I would buy the 8 GB version, not that it will be necessary.
The really nice way to do everything that needs doing is to use a Pi 400. Add a monitor and mouse and you're ready to go. I really like this platform. Everything to do with a project is running on the project machine itself.
I assume you will be recording to a file on the microSD card and some other process is going to read the file and present the output. You can add a printer to Ubuntu and print the data formatted by the program you write. Or, you can send the stuff over a network connection to a NAS or whatever.
Ubuntu gives you a lot of power.
There is a problem, however. There is only one accessible serial port. There are devices that connect over USB to provide a serial port. I used to use cable type converters. There are also some more expensive hubs but I don't know if either of these work on Ubuntu. You'll have to do some homework. Take a Pi USB port, run it into a USB hub and use USB<->Serial cables to the devices.
https://www.amazon.com/s?k=usb+serial+cableIf everything else fails, use a compute module at every external device and network the whole project to the host Pi. Berkeley Sockets is easy to use on Ubuntu. Use the compute module to provide a single serial port and grab the data for transmission over TCP/IP. I would probably start with this approach because it leads toward future proofing. Only the host code needs to change if I need to add 1 or 100 additional devices. Distance doesn't matter, I can record data from anywhere with an internet connection.