The employers have both committed offences - the first in not informing ImmD that your husband left their employment, and the second in employing someone illegally. But it is not the new employer's job to apply for a visa. Visa applications are made by the person concerned. The potential employer is the Sponsor of the application, not the applicant. You need to apply for permission to transfer sponsorship of an Employment Visa, and it needs to be granted before you can start working with the new employer.
Your point about taxes is valid and I guess they will understand there was no intent to deceive, but ignorance of the law is no defence, especially these days when all the information is so readily available online - the ImmD website is very clear and comprehensive.
Your husband has no option but to be very apologetic and grovel lots. If ImmD is convinced then you'll probably get away with it, but if not your husband (and the employers involved) could well be prosecuted, the result of which would probably be your husband being deported and the employers being fined. But this isn't very likely so long as your husband is suitably contrite with ImmD.