In Kate Atkinson's Life after Life, when the protagonist is born, her life is endangered (and in the first version she dies).  And the paragraph goes:

"Nothing. Slack and still, the little ribcage not moving. Sylvie's own heart was knocking in her chest as if a fist was inside her, punching its way out. Such danger! Like a terrible thrill, a tide washing through her. Instinctively, she placed her mouth over the baby's face, covering the little mouth and nose, She blew gently. And again. And again.

And the baby came back to life. It was that simple. ("I'm sure it was a coincidence," Dr Fellowes said, when told of this medical miracle. "It seems very unlikely that you could revive someone using that method.")

Would it really have been that unheard of to use mouth-to-mouth in 1910?  When did it come into common use?

I thought it was used at least as far back as the mid 19th century but it seems to have been around a lot longer.

This is from Wiki:

In 1773, physician William Hawes (1736–1808) began publicising the power of artificial respiration to resuscitate people who superficially appeared to have drowned. For a year he paid a reward out of his own pocket to any one bringing him a body rescued from the water within a reasonable time of immersion. Thomas Cogan, another English physician, who had become interested in the same subject during a stay at Amsterdam, where was instituted in 1767 a society for preservation of life from accidents in water, joined Hawes in his crusade. In the summer of 1774 Hawes and Cogan each brought fifteen friends to a meeting at the Chapter Coffee-house, St Paul's Churchyard, where they founded the Royal Humane Society as a campaigning group for first aid and resuscitation.

Gradually, branches of the Royal Humane Society were set up in other parts of the country, mainly in ports and coastal towns where the risk of drowning was high and by the end of the 19th century the society had upwards of 280 depots throughout the UK, supplied with life-saving apparatus. The earliest of these depots was the Receiving House in Hyde Park, on the north bank of the Serpentine, which was built in 1794 on a site granted by George III. Hyde Park was chosen because tens of thousands of people swam in the Serpentine in the summer and ice-skated in the winter. Boats and boatmen were kept to render aid to bathers, and in the winter ice-men were sent round to the different skating grounds in and around London. The society distributed money-rewards, medals, clasps and testimonials, to those who saved or attempted to save drowning people. It further recognized "all cases of exceptional bravery in rescuing or attempting to rescue persons from asphyxia in mines, wells, blasting furnaces, or in sewers where foul gas may endanger life."

It's highly likely that mothers and midwives etc have been using mouth-to-mouth resuscitation on newborn infants since the dawn of mankind. The ancient Romans, for example, believed that some animals such as lions were born nearly lifeless and that the breath of the father would then revive them. It's only a small step of imagination, therefore, to transfer the concept of 'life-giving breath' from fitful lion cubs to struggling human babies.

How widespread the practice was, however, is another question. In 1910, among qualified practitioners of Western medicine, the understanding of cardiopulmonary resuscitation (for infants and others) was still relatively primitive. There is some evidence that in Japan, the mouth-to-mouth resuscitation of adults has been practiced for centuries. In the West, the formal practice of cardiopulmonary resuscitation (cpr) dates, as Meles says, from the 18th century. Actual mouth-to-mouth of adults, however, only seems to have been officially adopted in the 1950s.

An excellent history of cardiopulmonary resuscitation can be read here:

http://www.nzats.co.nz/wp-content/uploads/Evolution-of-Cardiopulmonary-Resuscitation-.docx

And here is an academic study of neonatal resuscitation in the 20th century:

http://theses.gla.ac.uk/841/1/2008mcadamsphd.pdf