In my mind, the way to fix this is related to the fact that with a phrase like ne … que, you’re trying to “negate” a quantity or degree. The way that you currently have the sentence set up, you’re matching ne … que to the verb rather than to a quantity or degree. So, we can add that emphasis by trying to put a quantity or degree in there that we can then “negate” with ne … que

I would say something like:

Ce n’était qu’une semaine après son arrivée en France qu’elle y est allée …

That way, it’s the phrase “C’était une semaine après son arrivée” that we can “negate” rather than “Elle y est allée,” which doesn’t have a quantity or degree associated with it.

First, you do have the right suspicion :

Elle n’y est allée qu’une semaine après son arrivée en France.

can have both meanings :

She only went there after a week.

She went there after only a week.

It’s because ne…que pattern is, in this case, only mark of remarkable duration.
The meaning will depend on the context, like in these two examples:

• Remarkably long time :

  Il n’est venu me voir que 2 ans après son arrivée au village

• Remarkably short time :

  Il n’a passé son permis qu’un jour après son frère.

So, to ensure that your sentence is well understood when context is not obvious, I think you will have to add something to your sentence to make it unambiguous.

Here are examples to translate “she went there after only a week” without ambiguity :

• Adding seulement:

  Elle n’y est allée qu’une semaine seulement après son arrivée en France.

• Changing turn of phrase to add context:

  Il ne lui a fallu qu’une semaine pour y aller après son arrivée en France.

  Elle n’a attendu qu’une semaine pour y aller après son arrivée en France.

  Ça ne lui a pris qu’une semaine pour y aller après son arrivée en France

Ce n’est pas avant une semaine après son arrivée en France qu’elle y est allée.

Tac’s answer is right about the ambiguity, so you could say:

Elle n’y est allée qu’une petite semaine après son arrivée en France

Precising the week is little, you mean it was a short period. You could say courte (short) too.

Care about another ambiguity: the original phrase (and my proposition) could mean She only went there for a week, when she arrived in France.