Atmosphere Blowing Away
NASA estimate for Mars atmosphere stripping by solar wind:
about 100 grams/second
Earth is closer to the sun than Mars, so assuming a inverse square relation the solar wind would be about 2.5x stronger at Earth, Also earth's atmosphere is denser (160x) making a better target, but Earth has a higher gravity (3x) so my estimate would be for Earth to lose ~5 kg/s of atmosphere or 432,000 kg a day.
Earth's atmosphere is 5.15×10^18 kg
So at that rate of depletion the atmosphere would be gone in ~33 million years.
Of course some of the oceans gases would also start to evaporate at lower pressure and as the atmosphere decreased the rate would decrease so it would actually likely be longer, but as far as planets go that's pretty soon.
The big danger here is that the solar wind would blow away the ozone layer, which block ~98% of the medium-frequency ultraviolet light emitted by the sun, but as above it would not be instantaneous, and the rest of the atmosphere would continue to filter some of the UV radiation.
This would be bad, but unlikely a total extinction level event, greatly increased levels of skin cancers in humans and animals (easily +100x), and widespread biological damage with likely extinction of UV sensitive species (specifically bacteria). A lot of species have already evolved UV defense mechanisms, things like melanin, and actually actively use UV to produce vitamin D. With the increased UV levels life would adapt, humans may not survive the various collapses of ecologies, but life definitely would survive it.