The problem she has is that in the past it has been really easy to live in the UK with no legal status/no right to reside, and be able to steal from the NHS, work illegally, be allowed to stay in the UK even though they lied to enter the UK and said they were just a visitor, delay deportation for years, etc, but now all that abuse has ended. The two new Immigration Acts 2014 and 2016 and all the never ending changes to the Immigration Act 1971 and the changes to outside the immigration rules, are designed to make it impossible for those who don't have permission to be in the UK, to survive. Too much abuse always gets things shut down.
By overstaying, she has committed a criminal offence, not her husband. It is a fail of Good Character requirement to help an illegal and refusal of a visa, but if he already holds British citizenship then it won't affect him.
If her husband is renting a property or just a room in somebody’s' home, then she is not allowed to live there. A landlord must now check that illegals are not in their property even if that illegal does not have their name on a tenancy agreement. The government has an online checking system for landlords.
"You can get an unlimited fine or be sent to prison for renting your property to someone who isn’t allowed to stay in England."https://www.gov.uk/check-tenant-right-to-rent-documents/how-to-check It's not in any landlords interest to keep quiet about illegals now or to not bother to check. New laws are going through to allow landlords to evict illegals quickly.
Anyone who employs her is now committing a criminal offence.
Since 2016, working or driving while having no legal status in the UK, is also a criminal offence now too, with up to 12 months in jail and their wages/the car they were driving, seized. A UK driving licence is now revoked from those who don’t leave the UK and become overstayers, as UKVI now tell the DVLA if they have haven't left the UK.
No bank accounts for illegal overstayers anymore and the banks are now starting to look for those who already hold a bank account with them before this new law but have no legal status in the UK. The US is also looking for US citizens who have bank accounts in the UK and that too will bring her to the attention of UK banks.
Even if she can survive without: renting, using the NHS, driving, working; a bank account etc, there are still others ways she can bring herself to the attention of UKVI. i.e. the UK have exit checks now and will realise she hasn't left. Even if she moves, UKVI can now gather information from other government departments when they look for overstayers i.e. the NHS records, even if the person paid to use the NHS. The police will stop check on the occupants of cars too and now carry portable fingerprint machines for those they think might not be legal in the UK and will detain immigration criminals for UKVI; UKVI have immigration checks in the streets.