I have never seen a MCU that cannot write its own flash memory with a new program. This is how everybody makes firmware updates. At most, some MCUs have the option to blow some fuses to make some part of the memory immutable, but during program development the memory is still writable and used as such.

Ancient MCUs with true ROM or ultraviolet-erasable PROM may be claimed to not be a stored-program computer, but this claim is completely false about any MCU with a flash memory.

Manchester Baby was just a test device, not intended to be really useful, but its successor Manchester Mark 1, which became operational next year, in 1949, already had index registers.

Nevertheless, in the next few years there have been built many computers that lacked index registers, despite the precedent of Manchester Mark 1, so instruction modification during program execution was mandatory in them. After 1954, few, if any, computers remained without index registers, so instruction modification was no longer necessary.