An Orthodox conversion will be accepted by virtually everybody, as explained by DoubleAA. Non-Orthodox movements will tend to accept each others' conversions if they include all the ritual components, but Orthodox rabbis won't. So if your family wants to maximize acceptance and minimize challenges to the status of her and her future children, that's the safest way to go.

Given that analysis, one might wonder why anyone converts through other movements. I've seen a few reasons at play, the largest of which is that you aren't just accepting torah and joining a religion and a people, but you are also joining a local community. The convert must be comfortable with his rabbi (with whom he will have many deep, personal conversations) and with the congregation he will join. There are many factors that can affect one's choice of congregation and rabbi (beyond the scope of this question).

Another factor that can bear on the decision is family considerations. In the case brought in the question, the would-be convert is joining a Conservative family (who presumably belong to a Conservative synagogue). In addition to the community aspect, she may want to align her practice with that of her future family. While Conservative Judaism is a halachic movement, they understand some things differently from how the Orthodox understand them, and those differences could be enough to preclude an Orthodox conversion. (Of course, CYLOR.)

Finally, sometimes conversion is a gradual process. I've met people who converted first through one movement that they thought was right, then later changed to another and converted again. People are not static, and people coming in from the outside don't always "get it" all in one shot.

An Orthodox conversion will be accepted by virtually everybody, as explained by DoubleAA. Non-Orthodox movements will tend to accept each others' conversions if they include all the ritual components, but Orthodox rabbis won't. So if your family wants to maximize acceptance and minimize challenges to the status of her and her future children, that's the safest way to go.

Given that analysis, one might wonder why anyone converts through other movements. I've seen a few reasons at play, the largest of which is that you aren't just accepting torah and joining a religion and a people, but you are also joining a local community. The convert must be comfortable with his rabbi (with whom he will have many deep, personal conversations) and with the congregation he will join. There are many factors that can affect one's choice of congregation and rabbi (beyond the scope of this question).

Another factor that can bear on the decision is family considerations. In the case brought in the question, the would-be convert is joining a Conservative family (who presumably belong to a Conservative synagogue). In addition to the community aspect, she may want to align her practice with that of her future family. While Conservative Judaism is a halachic movement, they understand some things differently from how the Orthodox understand them, and those differences could be enough to preclude an Orthodox conversion. (Of course, CYLOR.)

Finally, sometimes conversion is a gradual process. I've met people who converted first through one movement that they thought was right, then later changed to another and converted again. People are not static, and people coming in from the outside don't always "get it" all in one shot.