Puppets have a -25% need reduction, but that's all. They have a -25%

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reduction and a -50%

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reduction, meaning they are a drag in the long run.
Unhappiness after conquest grow exponentially, your mistake was not recognizing its tipping point way earlier; it will take a long time to fix your happiness issue now, no matter what. There are few civs that can still manage unhappiness after lots of conquests with their uniques, and England isn't one of them.
Imperialism itself isn't wrong when playing England, since it works to your UA and UU advantages. Progress + Fealty makes for a strong infrastructure path, and you have an unique Factory to help; you should be able to fix this situation eventually. Imperialism also has a +25%

for puppets, which negates the puppet's penalty (they stack additively).
On a note, puppeting isn't much useful when going Fealty, since you can't buy monasteries in puppets. They would at least help you with the illiteracy problem, and you could focus on getting Castles and Armories in annexed cities, which are built faster and help dealing with unhappiness the most.
The only ways to immediately fix it would be Prora or Police State, which requires having the Autocracy Ideology. For now you'll need as many measures you can get:
- Select "avoid growth" in all unhappy cities; you can't have more unhappiness in a city than its population, it will at least prevent your cities from giving you more headache. Puppets may redirect some citizens away from food tiles to something more productive as well.
- Get a "We Love the King Day" immediately on all cities to make use of Fealty's 15% production bonus under WLTKD. You need infrastructure to counter this situation and you at least have picked two trees that are focused on that, make use of them.
- Redirect your spies to mercantile City-states and get some coups. Any extra Happiness you can get from a mercantile CS will help, and England has both an extra spy and the best spies out there. Mercantile CS also have an exclusive luxury under them that you can't find around, meaning extra happiness.
- Check for wonders that have a need reduction for your empire. Branderburg gate is an excellent one with its -30% Defense need in all the empire. Neuschwanstein is also powerful for your wide empire.
- If you have any Great Admirals, now's a good time to spend them for luxuries.
- Bulb any Great Writers/Artists/Musicians you get, their effects help a lot in this situation and you'll want to pick an Ideology ASAP for some of the most powerful happiness and need reduction policies of the game.