Actually, I think it is doable.
To go to all 3, I would stay in Santa Fe at the Santa Fe Skies RV park.
Easy drive on Turquoise Trail to Albuquerque and back in 1 day.
Drive to Taos another day take the High Road to Taos one way. I think it is about 1 1/2 hours drive the normal way on NM 68. Longer drive on the High Road.
Abiquiu another day, about 1 1/2 hours drive each way.
I would drive a loop around Valle Caldera. Take I25 to US 550 to NM 4 back to US84. About 4 hours driving. And maybe Bandelier National Monument same day or another day. Depending on how much walking you and your mother can do.
And of course shopping in Santa Fe.
We have done all these trips as day trips. So it will kind of depend on how much driving you would want to do.
Weather will usually be hot and dry in June, earlier will be less hot.
The lower elevations maybe up to the upper 90's. The higher elevations upper 80's. We are at 7300' and our record high is 95 which it will only get to if the humidity is down to 5% and usually won't get above 90 unless the monsoon is late. Average start is July 4th. Actually, we got a little snow the beginning of June last year I think it was.
David