The MMU isn't part of this attack, and there are ARM cores with MMUs that are not affected. Cortex M cores are in-order CPUs, so they do not have branch prediction or perform speculative execution which the attacks rely on.
The attack is all about bypassing memory protection, which without an MMU to provide any, is moot --- code has access to all of memory to read normally anyway.
Cortex-M has an MPU, not MMU to provide protection for memory regions.
But as said above, the MPU/MMU has no effect on this bug, this is ab out speculative execution and determining information from a cache side channel attack.