just a reminder to make sure that the travel insurance policy you say is 'covered' allows you to work, and a reminder that it doesn't cover you if you are drunk.
Got point neverwinter although it does raise a whole new can of worms for CarlyAnn to consider.
First there is the question of how long can they be out of the UK and still have NHS coverage if they return to the UK. There is a time period, I don't know what it is. If you are gone over that period, when you return you have to re-register for NHS and there is a period before coverage begins. 3-6 months?
Second, as EU residents they can get a card that gives them emergency treatment in all EU countries while travelling. http://www.nhs.uk/NHSEngland/Healthcareabroad/EHIC/Pages/about-the-ehic.aspx But that is not 'travel insurance', just emergency medical coverage. I believe it is for one year and can be extended a second year. But it assumes you are just a tourist or are working legally. http://www.nhs.uk/NHSEngland/Healthcareabroad/movingabroad/Pages/Workingabroad.aspx
Third, no private insurance policy will cover anyone who is drunk (and has an accident) or is doing anything illegal. Illegal includes working 'under the table'. So a seasonal bar worker for example would void their insurance coverage if the job is paid 'under the table', which most are.
Fourth, yes, not all travel insurance will cover someone who is working even it they are legally working. So that has to be checked.
Regarding the 'drunk' issue. It used to be that any tourist who went to Emergency in a Greek hospital was treated for free EXCEPT in the case of someone who was obviously drunk. They were made to pay. I don't know if that is still the case given the current economy in Greece and the austerity measures but it is worth knowing how things may be viewed.
So bottom line, you need to know you are covered somehow CarlyAnn and nothing will cover you if working illegally.

the NHS time limit is interesting. You are supposed to tell your doctor if you leave for more than three months, and de-register. I cannot imagine that anyone who goes on long trips ever does that, and I can't say I blame them!
most UK travel insurers will cover for long periods, and so must be assuming that the NHS will pick up in case of repatriation. Lack of joined up thinking there.
this is separate from the work and drinking issues which are entirely valid. OP, please note.
Yes the NHS and the British government has been quite lax in the past about paying attention to how long someone has been out of the country. As a Brit who emmigrated to Canada I can well remember other expat friends of our family who used to fly to the UK for a visit with family and see a dentist while they were there. In Canada you pay for dental treatment and in the UK they could get it for free.
However, those days are gone I believe. Now they do a bit more checking. Answers are here: http://www.essentialhealthltd.co.uk/international/international-considerations/who-exactly-is-entitled-to-access-the-u.k.-nhs.html
Basically you can lose your coverage after an absence of 3 months if you have MOVED abroad. So someone who is on say 1 or 2 years of extended travel but still maintains their residency in the UK is OK. Move to Canada or Australia (for example) and you lose it in 3 months. It seems the qualifier is 'moved' vs. temporarily out of the country. They still seem to be pretty easy going in terms of long term travel. Canada for example will cut off snowbirds who go south to Florida for the winter for more than 183 days exactly.
So let's say you show up with dengue fever. The GP isn't stupid, s/he knows you don't contract that in the UK. So he is required to check on your residency and eligibility. Caught. You're caught because it is likely to come out you have been living/working somewhere else, not just travelling.