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The prefix "Eil-" is related to the verb "eilen" - to rush, to hurry, to hasten. In a compound Eil-X means that X is either very fast moving/working/progressing itself or that X has to be done very fast. "Eil-" is pretty common and can be prefixed to a lot of words so that is not possible to have an entry in a dictionary for every Eil-compound. So, an "Eilgesetz" is a law that is to be adopted very quickly (sometimes quickly enough that the vast majority does not even notice the adoption of the law).
Your derivation of the root is incorrect. "Eilgesetzen" is the dative plural of "Eilgesetz".
You will probably already find translations for this, but it can be decomposed further. In fact this is a compound of "Eile" and "Gesetz". The last e of "Eile" is lost in concatenation.
These two are the root forms. Translating them yields "haste" and "law". If we now put that together with the notion of germans to include adjectives as a compound to a noun, we get "hasty laws" as a translation.
If you want to keep it a compound though I'd recommend "express-law"
The whole sentence then becomes: "The government in London wants to change regulations on the surveillance of telephone- and internetconnections by means of express-laws"
