Florall, I woudn't put too much store by what Bill Oddie says - he's a comedian and a TV presenter, not an ornithologist or ecologist or authority on bird behaviour. Remember, he's basically a birdwatcher with no more science experience than the average bloke in a hide. He studied English at university, not Zoology.
The second pair aren't using the oak tree as that is in the first pair's territory, so they don't have access. The kind of thing you describe is common with suburban/urban Blue tit territories, as the birds are at artificially high densities due to the good artificial food supply in winter. The problem is that there isn't enough natural food to rear chicks. This is exactly the kind of situation I've got in my box at home, with the adults foraging in just 3 birch trees and a line of rose shrubs - basically all they have is aphids to feed the young on. They've started taking in sunflower hearts and also lard, but, as yu figured, the moisture content of these is so low that the chicks will die of dehydration if they start maxing out on it. Especially if the weather turns bad.
What Bill Oddie thought was 'inexperience' or 'bad parenting' on Springwatch last year, where the female kept placing a sunflower heart in a chick's gape, then taking it out for another one etc, is perfectly normal behaviour if you think about what they do with caterpillars - they squish them up and give each chick a little glug of the juice, so they all get a bit of food/moisture. That's clearly not going to work with sunflower hearts, but if it's all the blue tits have in their territory then they try it out of desperation. Blue tit territories of 1 pair per tennis court area works fine in woodland, as they all have trees dripping with caterpillars right outside the nest, but it doesn't work in suburbia where non-native trees/shrubs often don't have much food on them. They can keep 2 birds going throughout the year, but not an additonal 10 mouths in spring.
The only thing you can do, if you want to try and help, is to provide mini-mealworms, but this is expensive and will probably be raided by every passing starling and blackbird too. At the end of the day, your 2nd pair are rolling the dice and they may or may not bring off some young. Don't be too disappointed if they don't, it's all part of the checks and balances and just shows that your neck of the woods is super-saturated with blue tits - it's a sign of how well they're doing in what is a marginal habitat for them. Think of them as a reserve pair, in case anything happens to the birds with the good oak tree territory.