I'll try and help answer those questions regarding London.
When you enter, I would recommend heading directly to one of the zoo's three big buildings, Tiny Giants, Blackburn Pavilion and the Clore (labelled on the map as 'Night Life' and 'Rainforest Life'). Tiny Giants is amazing, possibly the greatest invertebrate house in Europe, but it is difficult to appreciate the later that you leave it due to crowds. However if you are not interested in invertebrates, then you don't have to prioritise it. Blackburn Pavilion is a nice bird house in a Victorian building, with a nice atmosphere that is also better enjoyed when the building is mostly empty. There are also a few nice rarities, if you are interested in those, such as Scarlet-chested Sunbird in the main walkthrough, and Splendid Sunbird and Collared Trogon in the smaller, second walkthrough. The Clore isn't a very high quality enclosure, but it is packed with rarities, such as Potto, Moholi Galago, Grey Slender Loris, nominate Southern Tamandua and Narrow-striped Boky.
Regardless of when you chose to visit the Clore, I would recommend you enter through the Lemur Walkthrough, which finishes in the building. This is because the walkthrough is one-way, and is thus the only way to see the indoor holdings for the lemurs which is home to the zoo's Aye-Aye! This takes you out into the main lobby where you can chose to visit either the nocturnal or rainforest section first. If the Clore is the first exhibit that you visited, then prioritise the nocturnal animals (on the lower level) to evade crowds. If not, then it really doesn't matter. There are a lot of nice species in the Clore, with Potto and Grey Slender Loris being especially worthwhile.
Monkey Valley is the zoo's newest exhibit - an impressive renovation of the historical Snowdon Aviary that serves as a Colobus walkthrough. It is a unique experience, but the queues are absurd, you are guided by keepers when inside the enclosure, and the walkthrough is closed for the last two hours of the day so that the monkeys can sleep. I would recommend checking the enclosure, as it *is* amazing, but if the queues are too long, then it is probably not worth it...
If you want to see the zoo's Sumatran Tiger cubs, then it is unlikely (although possible) that you will see them from the canopy viewing area. Instead, go up onto the boardwalk towards the zoo's gibbon enclosure, where there is a raised viewing of their secondary habitat. The cubs are more often seen here, especially climbing on the rocks that line the back of the enclosure.
As for what areas to skip?
Well, for starters, there isn't really anything worth seeing in either of the zoo's two children's zoos (Animal Adventure and the Ambika Paul Children's Zoo). Personally speaking, I also recommend skipping Meet the Monkeys (a squirrel monkey walkthrough) and the Komodo Dragon House, as recent renovations have left them as inconvenient dead ends, and there is little that stands out. The Reptile House is closing soon, and as such many species have already left, but there are still nice species in there and its impending closure only makes seeing it more worthwhile in my opinion. However, it is really your choice as to what is worth skipping.
I hope this helps!