Hello there. Sorry that you didn't get a reply earlier.
Provided the parents are pure on their e-loci, the father for Wheaten, the mother for Birchen, you will get Birchen with extra leakage (from the Wheaten making itself known); each offspring will have one copy of each gene. If the mother is pure for Mahogany (which gives her neck the lovely dark red colour) then the leakage will be reddish, though maybe not as deep as her own. Around half the chicks will have blue, however, as the mother is the only one providing the lacing genes and melanisers required to produce a good laced blue, they will likely lack the nice black lacing your hen ought to have.
If you would share pictures of the parents (and chicks if you'd like), we can make a better assessment of their genetics.