It is permissible for the wife to wash and shroud her deceased husband and the husband to wash and shroud his deceased wife.

The wife washes her deceased husband and the husband washes his deceased wife.

عَنْ عَائِشَةَ، قَالَتْ رَجَعَ رَسُولُ اللَّهِ ـ صلى الله عليه وسلم ـ مِنَ الْبَقِيعِ فَوَجَدَنِي وَأَنَا أَجِدُ صُدَاعًا فِي رَأْسِي وَأَنَا أَقُولُ وَارَأْسَاهُ فَقَالَ ‏”‏ بَلْ أَنَا يَا عَائِشَةُ وَارَأْسَاهُ ‏”‏ ‏.‏ ثُمَّ قَالَ ‏”‏ مَا ضَرَّكِ لَوْ مِتِّ قَبْلِي فَقُمْتُ عَلَيْكِ فَغَسَّلْتُكِ وَكَفَّنْتُكِ وَصَلَّيْتُ عَلَيْكِ وَدَفَنْتُكِ ‏”‏ ‏.‏

From ʿAishah: “The Messenger of Allah (ﷺ) came back from Al-Baqi’ and I had a headache and was saying: ‘O my head!’ He said: ‘Rather, I should say, O my head, O ‘A’ishah!’

Then he said: ‘It will not harm you if you were to die before me, for I will take care of you, wash you, shroud you, offer the funeral prayer for you and bury you.’”
(Ahmad, Ibn Mājah, no. 1465, Ad-Darimi and others — graded saheeh by Al-Albani)

كَانَتْ عَائِشَةُ تَقُولُ لَوِ اسْتَقْبَلْتُ مِنْ أَمْرِي مَا اسْتَدْبَرْتُ مَا غَسَّلَهُ إِلاَّ نِسَاؤُهُ

‘A’ishah used to say: “If I had known beforehand about my affair what I found out later, none would have washed him (i.e., the prophet) except his wives.” (Abu Dawood, 3141, Al-Hakim and others — authentic)

Ref: See Ahkām al-Janā’iz of Al-Imām Al-Albānī (rahimahullāh)