I speedrun some N64 games.
I'd say the most likely outcome for legality of the MiSTer core in speedruns is that most game communities won't explicitly allow it, at the very least not as a competitor to original N64 hardware. This is especially true for RTA (real-time attack) speedruns, or runs where final time is the real time elapsed during a single-segment run. This does not preclude FPGA runs being allowed under "emulator" categories, though, which already include runs done on somewhat-inaccurate emulators, most commonly in that they don't perfectly reproduce the "lag" (framedrops, etc.) of an N64 console.
However, there are many games that utilize an in-game timer (Wave Race 64, Mario Kart 64, etc.). These are seemingly riper candidates for full legality, as you don't save any time due to reduced lag on the core over real hardware.
Ultimately, though, every single speedrun community is a unique entity. Decisions made in one game do not carry over to any others. This is also usually not something that happens overnight, as speedrun moderators are often loathe to allow new things without extensive testing, and the MiSTer core is still relatively new, not everyone has MiSTer hardware, etc.
Things to think about.