I've got a new cough, does that now mean

[MissFrizzleIAmNot]

I have to take my kids out of school for 14 days including my 15 year old?

Yes, entire household.

Yes and if they come down with it, the 14 days restart

Yes yes yes

Yes. And self isolate - which means not really leaving the house.

So if in 6 days, say my DD starts coughing then it's 14 days again from then? For everyone?

For the people who have not had symptoms yes.

If you were the original ‘cougher’ you would be ok

Will the schools advise on what work to do with the kids??

But what if op's original cough was just a cough and not Coronavirus, how do you know when to stop self isolating?

Yes and if they come down with it, the 14 days restart

Reading the government guidelines, this is not correct. The 14 days don't restart

If you come down with it, you move to a 7 day isolation, while everyone else remains on the same 14 day isolation

Should a household member develop coronavirus symptoms late in the 14-day household-isolation period (eg on day 13 or day 14) the isolation period does not need to be extended, but the person with the new symptoms has to stay at home for 7 days.

This is from the government guidelines. Doesn't seem totally logical but there you are

Yes, it is a bit confusing, but see this link:

www.gov.uk/government/publications/covid-19-guidance-on-social-distancing-and-for-vulnerable-people/guidance-on-social-distancing-for-everyone-in-the-uk-and-protecting-older-people-and-vulnerable-adults

which has a link to:

www.gov.uk/government/publications/covid-19-stay-at-home-guidance

If you live alone and you have symptoms of coronavirus illness (COVID-19), however mild, stay at home for 7 days from when your symptoms started. (See ending isolation section for more information)
if you live with others and you or one of them have symptoms of coronavirus, then all household members must stay at home and not leave the house for 14 days. The 14-day period starts from the day when the first person in the house became ill
it is likely that people living within a household will infect each other or be infected already. Staying at home for 14 days will greatly reduce the overall amount of infection the household could pass on to others in the community
for anyone in the household who starts displaying symptoms, they need to stay at home for 7 days from when the symptoms appeared, regardless of what day they are on in the original 14-day isolation period. (See ending isolation section for more information)