The cheap and easy solution is to buy an Instek GDS-2072E and hack it. It's everything you need. I've set up a Zynq/Cyclone dev platform to implement FOSS FW for COTS DSOs. There's a thread on that. At the moment I'm rewiring my workspace. I bought my 2072E (now at 300 MHz) as a test bed with the good luck to get it for $244 from Amazon. However, even at the regular price, it's a good platform. I think Siglent also sell a Zynq based DSO.
ALL the low cost DSOs are FPGA based so anyone of those will work. The Zynq is particularly attractive because the boot is public stated.
If you want "fast" as in 2-3 GHz BW, you might want to consider buying a LeCroy DDA-120 and the FPGA board of your choice. Replace the CRT with a $90 RPi display and use all that space to put proper anti-alias filters. A single channel at 4 GHz Nyquist is fast! I just bought a DDA-125 which already has the LCD display. I think now that a 120 and a 10" VGA LCD is a better option.
There are two lines which have both an FPGA fabric and hard CPUs, the Zynq which is already in wide use for this application, and the Cyclone V.
Probably most of the chip vendors have reference designs for AFEs for DSOs. If you want *really* fast, then there are ADC dev boards for a few $K which will give phenomenal results.