It was neck and neck, in my opinion. In the UK, ARM ruled, but in the US, the MIPS Magnum series were well received and widely used. MIPS had a chance to be an ARM, but it was squandered.
Just guessing, but it could have something to do with ARM not doing actual hardware. Instead they license either chip designs or the ISA to anyone willing to pay.
In a way it is reminiscent of the z80 clones that was all over the place in the 80s.
I have always wondered how ARM succeeded when MIPS was already around. Anyone know ?