What do these slang phrases in Dahl's "The BFG" mean?

Solution 1:

You are correct about "not on your nelly":

not on your nelly
(idiomatic, Cockney rhyming slang) not on your life, an emphatic form of no.

(Wiktionary)

Nelly Duff: (UK) Nelly Duff = puff (breath) 'Not on your Nelly' is used to mean 'Not on your life'.

(english-for-students.com)

As you can see, "nelly" is an abbreviated "nelly duff" (breath), and breath by extension means life. Hence, "not on your life".

I'm not sure about "pop goes the weasel" in this context, but let me guesstimate that contextually this is roughly equivalent to "all hell will break loose".

If all hell breaks loose, a situation suddenly becomes noisy and violent, usually with a lot of people arguing or fighting

(FreeDictionary.com)

In the context, the BFG doesn't want to wake the giants, or noisy violence would apparently ensue.

Solution 2:

Up and down the city road
in and out the eagle,
thats the way the money goes...
Pop! Goes the weasel.

In cloth making, the machine that wound the yarn was called the weasel. Every 1,000 yards, the machine made a popping sound, thus "Pop! goes the weasel." (lifted from wikipedia.) The phrase suggests that all has, or will go wrong, up and down the city road in and out the eagle...the eagle and child was a pub. So repeated visits may have led to a lack of clarity etc therefore Pop goes..