I've lived in Singapore on two occasions for a total of 8 years.....loved it there but the rules and regulations are very strict indeed. When the landing cards are issued to anyone arriving in Singapore it is clearly marked in big red letters "Death to Drug Traffickers', they mean it.
If you want to become a Singapore citizen you need to prove that you have given up your other citizenship ie. For Brits you would need to formally renounce British citizenship in front of a British Consular official (there isn't an Embassy as Singapore is a Commonwealth country, so there is the British High Commission building in Tanglin Road). I presume at this point they retain your British passport and hand you a letter or certificate of some kind proving this.
Btw, Singapore is quite choosy on whom they will allow to become a Singaporean. There are about 215,000 maids in Singapore from places like the Philippines, Sri Lanka, Indonesia etc. They could be working in Singapore for many years but are not allowed to apply. The country is tiny (the size of the Isle of Wight) with 4 million people living on it and no resources.....they only want people who can contribute by paying a lot into the tax system. There is no social security provision in Singapore, families are expected to pitch in if someone is unable to work, so low earners are not enouraged to stay indefinitely. Maids aren't even allowed to marry Singaporeans or have babies in Singapore....(the locals seem to frown on they kinfolk marrying maids, but I did know one maid who went home to Manila to marry a Singaporean guy and they had to jump through some hoops to move back to Singapore).
I too think this lady may have wanted to 'have her cake and eat it'. I know a few Singaporeans who went to the US and Canada for university and stayed on to work and eventually get married after their studies. One has a US passport, the other a Canadian passport (she married a German guy in Canada and they currently live in Switzerland). They have both retained their Singapore passports - but just keep it quiet to the authorities there.
This lady as far as I can tell wanted to keep ownership of property in Singapore (for example only locals can buy land or freehold landed property and this is why she wanted to retain residency for so long in Singapore. HDB flats - Housing Development Board - are only for citizens and permanent residents as they are subsidized by the government. Foreigners have restrictions on what they are allowed to buy and have to buy at full market rates).
I also suspect that her husband returned to live in England with their sons because sons of Singapore citizens and sons of permanent residents must register for National Service and conscription is mandatory from the age of 16.5 (can be deferred if still at school or college) for 2 years of full time NS. They still have to be available up to the age of 40 (my dentist had to delay my treatment as he had to go off and do a fortnight of NS and he was in his late 30s!). There are huge fines (and possibly imprisonment) if NS is not completed. The only way to get out of it if you don't want to pay a huge bond - which is forfeited if the son doesn't return to do his NS - is to leave Singapore. I am pretty sure that the spouse chose not to go for Permanent Residency because of this and perhaps if his work contract ended - along with his employment pass - instead of looking for another job in Singapore, the couple decided to move the boys to England whilst she stayed on in Singapore to keep the property there (property is generally more expensive than London prices, even for a tiny flat).