Most women who gave their profession as 'Midwife' or 'District Nurse' in 1901 had received no formal training, and learnt their craft from watching others, and then gaining 'on the job' experience.. There was no regulation of training or registration of nurses, so anyone could call themselves 'nurse' or 'midwife', and as long as they worked responsibly, and within the law, they were free to practise as they wished.
The Central Midwives Board was set up following the Midwives Act of 1902, and Registers of Midwives published from 1904. Most of the women applying to be registered had no formal training, so were accepted onto the Register according to their experience and references - they had to prove they were safe. If your grandmother was still working in 1904, then it is possible that her name appears in the Registers, but many thousands of midwives did not register, often because they could not provide satisfactory evidence of skill. The Registers of the CMB are now kept at The National Archives in class DV7.
Women often did a combined job of nurse/midwife, particularly in rural areas - there had been village 'wise women' since time began, but it was not until the Midwives' Act [1902] and later the Nurses Registration Act [1919] that any lists of these were kept - the vast majority of pre-1919 nurses cannot be traced in official records.
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