This is grim news. Thank you for your answer Sirius.
I have just received word from HMRC that my wife does not have the right to reside in the UK, and now I am wondering if this applies to my children as well.
Man, talk about having the rug pulled out from under me.
Are your children in the UK on the UK Tier 2 dependant visas? If they are then they have permission to reside in the UK while your visa is valid, to attend UK state schools and use the NHS for free.
If they entered on German or US passports then they are visiting. The German passport will give them 3 months and the US one will have a date stamped in it.
Thousands of EEAs and their dependants are in the UK when they don't have a right to reside and the UK is getting tough on these because they cause a problem with UK schools, housing, welfare, NHS, jobs, services, roads ect. The UK are are talking about deporting but at the moment they starting by removing all benefits including housing and child related benefit, no matter how many years they have been claiming and stopping them from using the NHS for free.
From 2015, their new systems will be able to check and bill for the NHS (it was too easy to abuse the NHS before) if the EEA citizen is not a worker qualified person (earning enough each week to pay NICs) or on UK JSA benfits (now limited to 3? months).
This NHS checking that is now coming in is why you urgently need to sort out full health insurance for all your family that aren't on a Tier 2 dependant visas. The UK won't care if you pay for their treatment and medication using a German EHICs to pay as long as the UK doesn't pay, although you will have to pay for anything the German EHIC won't cover.
At least you are now aware that your wife (and children?) don't have a right to reside. Some non-EUs citizens seem to think that if they arrive in the UK with a EEA citizen that their EEA permit then allows them to reside in the UK for years, little realising that if their EEA citizen is out of work and job seeking in the UK for too long, then they and their dependants lost their right to reside. With the new NHS checking by looking to see if the EEA citizen is in work in the UK, the first they might know about it is when they are told they will have to pay for their healthcare. Before the NHS checking, the first they knew about it was when they thought they had got Permanent Residence in the UK and applied for it, only to be refused.
The new Exit checks and Biometric Residence Permits are going to be making it easier to block those who don't have legal status to be in the UK.