Is an expiatory oath (kaffarat yamin) incumbent upon a mother if she swore to go to the doctor in the first week of March, without specifying a particular day, but then was unable to go until after March 20th due to her health conditions and adverse weather?

The first of March refers to the first day, and it may also refer to the first few days of the month. However, anything after mid-month is not included in that definition, neither customarily nor linguistically. Therefore, your mother has broken her oath, and her illness and the bad weather conditions do not prevent the breaking of the oath.