As you suspected, in general it's not realistic that a duck would raise a swan, especially in the wild.
A swan's egg is much bigger than a mallard's, so the mother mallard (hen) is not physically equipped to incubate it. She can't cover it enough to heat it, and isn't strong enough to turn it over for even warmth. It also raises her up off the ground so her own eggs don't get the covering they need.
As the last egg is laid, the female mallard starts to incubate. She sits very tightly, and her brown plumage blends her perfectly to the background. Source
The incubation period is different, with the swan's being longer. Mallard eggs are incubated for approximately 28 days, swan eggs between 35 and 42. Once the hen takes off with her ducklings, the swan eggs would be left alone. The cygnets either wouldn't hatch or would die after birth. The mallard hen almost always incubates her eggs alone. With the swans the duties are shared, including both the pen (female), and the cob (male.)
The duck eggs only need one parent, while the swans need both. The drake (male duck) leaves the nest shortly after it's built, and the hen incubates on her own. The swans are incubated by both parents.
Important imprinting communication takes place during the embryonic phase, and it's different for the mallard than the swan. The ducklings communicate with the hen, and they initiate it. The cygnets communicate with both parents, and the parents initiate it.
Shortly before hatching, the nearly developed embryos produce clicking and peeping sounds inside the eggs. Known as "pipping," these vocalizations help synchronize hatching among different members of the brood. The hen responds by making soft clucking sounds to her unhatched ducklings. These early communications are a crucial part of a process known as imprinting, in which the ducklings learn to recognize their mother's voice. This ensures that the brood will follow their mother when it's time to leave the nest.
Even from quite early on in the incubation period, the developing embryo will be able to hear sounds inside the egg from the outside world. It is believed that the pen (female) and cob (male) make a number of sounds during the nesting period that they use to start the imprinting process of their offspring.
After the cygnet is born, the mother and father make a number of sounds the baby swans use to programme themselves to audibly recognise their parents. Each swan produces its own unique sound, rather like humans have their own unique voice. The pen has a slightly higher pitched sound than the cob.
The hen leads the ducklings to the water about 10 hours after hatching. They can swim and feed themselves, but she needs to teach them what to eat. The down of newly hatched mallards is not waterproof, so the hen warms them with her waterproof down for a few days until they grow their own. She also protects them from other ducklings. Ducks do not tolerate stray ducklings close to their own brood, and females kill small strange young they encounter.
The cygnets spend many more hours in the nest than the ducklings, and both the peg and the cob take them to the water. It’s on their second day that the family will take their first all important swim. As with ducks, the cygnets need the parents to keep them warm in the water while they grow the proper plumage. However, young cygnets will get tired of swimming and climb on the pen or cob's back for a rest. (See picture below) Although usually about 3 weeks, this can last up to 3 months. Ducks don't do this. Even if a hen tried to help a cygnet, it would be too big and heavy for the her, and it may fall off and drown.
Ducklings stay with the hen for at least 2 months before they can fly. During this time she's very protective.
She also protects her ducklings from attacks by other mallards. Ducks do not tolerate stray ducklings close to their own brood, and females kill small strange young they encounter. Source
Cygnets stay with their families for a longer period.
Once the cygnets hatch, swan parents live with their young for six to nine months. Source
After the mallard ducklings have flown, the hen wouldn't assume the responsibility for the cygnet for those extra months.
With all these factors in play, you can see why the story was just that, a story. Interestingly, I spoke at length with the Director of the Trumpeter Swan Society, a conservation organization in Minnesota. This is one of the most common questions she gets, because people who've read the book want to know if it could be true. She said she uses these calls as a way to get people excited about swans and ducks, to encourage conservation efforts.
She did say that in rare cases, but only in captivity and with a tremendous amount of work on the part of breeders or people who raise these birds, it has been successful. However, it's not the same as trying to it with chicken and geese, which can work because they're closer in physical and behavioral traits.
Swan pen and cob giving tired young cygnets a rest:
Sources for information referenced here and for additional information:
Royal Society for the Protection of Birds (RSPB) Mallard Life
Waterfowl Conservation Center-Mallards
Cornell Ornithology, All About Mallards
Cornell Ornithology, All About Swans
Trumpeter Swan Society