Depending on what you can afford, I would think you should consider a live-in carer, especially as it's only for a few months. You'll get more for your money than paying hourly rates, and you'll have peace of mind knowing that your mother's care and your child's care will not be in conflict.
I pay £85 per 24 hours for a live-in carer, plus agency fees, plus travel costs, plus bed and board. The carer has her own bedroom (and own bathroom, but this is not absolutely essential). The carer obviously needs personal time, and sleep, but in your situation you can cover the 24 hours between you. The bonus is that you could share cooking, shopping and laundry duties - by negotiation.
If you find that your mother needs support from the carer at night you'll be charged extra, but you could choose to be 'on duty' yourself instead. If your mother is very restless at night and sleeps in the day, then you can negotiate with the agency to adjust the carer's working hours to be 'waking nights', with the carer sleeping in the daytime, when you would look after your mother. I mention all that to show that with live-in care you arrange a care plan that suits your family's individual needs, and can make adjustments quite swiftly if things turn out differently than you expected.