I honestly don't think you will have a problem with Access America. As long as your mother being in the hospital is a covered condition (depending on how they define family member) and you bought the policy last August, they will not try to come back and say she could have had the obstruction back then and was just waiting for the trip for the surgery.

They may require some documentation from the doctor or hosiptal, they should be able to tell you up front what you will need, but I don't expect it to be that difficult for them to pay out what the policy says they will pay.

Beware, most travel insurance policies have things like how quickly you need to cancel once the hospitalization or whatever occurs, in some of it's policies Access America states:

For all of the covered reasons outlined below, You must notify the appropriate travel supplier(s) of Your cancellation or interruption within 72 hours of the occurrence, unless the condition prevents it, then as soon as reasonably possible. Otherwise the right to compensation will lapse.

You should check your policy to see what the time limits, if any, are and make sure to cancel within them.