HDMI and DVI-D are a digital representation of a tv signal --- there's all the blanking and porches and stuff (audio data is often interleaved into the blanking periods). You could process that in realtime, even though most displays don't (including CRT HDTVs; I trust your maybe one doesn't).
The latency incurred for upscaling depends on number of the vertical filter taps and the horizontal scan time. We’re talking order of 10 microseconds.
The only exception is if you’re using a super fancy machine learning whole-frame upscaling algorithm, but that’s not something you’ll find in an old CRT.
I've done it in Verilog on FPGA, for example.
But we do so on our machines because that's how the GPU/display controller pipeline works easiest from a software POV. That latency would be present on a CRT as well. What would be missing is the internal latency present in some monitors.
still, it's usually only 2 or 3 frames in game mode on a good display
This is a thing in the last 5-10 years, but was absolutely not a concern to the majority of display manufacturers before gamers complained. 100ms is significant in twitch games where network latencies are 20ms or lower.
Most modern systems (and computers) are sending digital signals where entire frames need to be constructed and reconstructed in a buffer before they can be displayed. Many HD CRT televisions have terrible latency because they’ll take the analog scan line signal, buffer it into a frame, scale it to the native CRT resolution, then scan it back out to the screen. A high end PVM might allow a straight path, but there is maybe one Toshiba HD CRT that doesn’t introduce several frames of latency (iirc).
That said, from 1999 to 2008 I ran 1600x1200 on 19in CRTs and except for professional LCDs, nothing had resolution, pitch, and color that came close. For 2008 was the inflection point where cost and quality of LCDs exceeded CRTs.