Is it permissable for a Christian wife of a widowed Muslim with children to adopt the children as if they are her own? I have been reading about adoption in Islam and can't see an answer to a situation such as this.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Adopting children is absolutely forbidden in Islam, so it is not permissible for this woman to adopt those children because she is not really their mother; Allaah Says (what means): {And He has not made your claimed [i.e. adopted] sons your [true] sons.  That is [merely] your saying by your mouths, but Allaah Says the truth, and He guides to the [right] way.}[Quran 33:4]

Therefore, the case that is mentioned in the question is not an exception to that general ruling.

For more benefit, please refer to Fataawa 84422 and 82371.

Allaah Knows best.