I had a very similar problem - A working A1200T boxed and stored in 2005 and unboxed in 2015. Same specifications, except an Apollo 1240 rather than a Blizzard, and a somewhat troublesome 2B motherboard.
It booted sometimes, if I got a black screen and endlessly-blinking power light, giving it a jiggle got it working temporarily. I'm not sure exactly what fixed it, but here are things I did...
First, I created a bracket from Plastimake to keep the accelerator stable. An improvement, but not much. I cleaned the contacts on the edge connectors and carefully prised some of the pins of the Apollo card out a bit so they had better contact and cleaned them as well. I still had problems so reseated the SIMM a few times, which had the same result as wiggling the card.
Then I re-arranged the cards - fast ethernet in slot 1, Voodoo 3 in slot 2, SB128 in slot 4. Set the jumpers to CONFIG and 8MB memory window, appropriate for a card with > 8MB memory.
Software-wise, the Mediator MMU variable is ON and VoodooMem is 15. (I think this leaves a 1MB "buffer" for sound and networking?) and I replaced SETPATCH with RemAPollo. I think that made the biggest difference - It probably initialises the MMU and CPU better than setpatch because it's a patched 68040.library.
For good measure I did the Rev2B Ian Steadman "timing fixes" but that was not really necessary although some old games worked better.
What I have now is a very stable system - I had it running nonstop for several days with Quake timedemos. If I could hazard a guess at the failure in the Mediator I would probably say the MMU (or the way it's initialised) or possibly faulty RAM on the Blizzard card could be the culprit. It took me months to sort out, though, so best of luck!