Mid 2011 iMac 27'' won't boot - 3 beeps and maxed fans
Upon pressing the power button, I’m greeted with 3 beeps every 5
seconds. After around 35 seconds all the fans are maxing out on top
speed.

There's a RAM issue.
3 successive beeps followed by a five second delay, then repeating means &quot;RAM didn't pass an integrity check.  Apple has an excellent support document that details the startup sounds:  About Mac startup tones
Normally, this would be limited to the...

RAM Modules
Memory slots (logic board)

To diagnose this, you want to remove all of the memory and try each module in each slot individually.  The idea isolate whether it's a memory module problem or an issue with the slots on the logic board.  You should get one of the following results:

If one or more modules gives you an error in every slot, it's that/those modules.

If all modules give a problem in one or more slots, it's that slot
(Draft)


If you can't narrow it down to one of those two possibilities, there's a &quot;larger&quot; issue that needs to be investigated.

Cause:  CPU Issue

I had removed it [CPU] the first time round to repaste it. The thought process was that the iMac was almost 10 years old and had never been repasted.

 
In this case, this is where the issue occurred.  Intel CPUs directly connect to a number of peripherals like display and memory.  (A memory buffer actually, but for all intents and purposes, it's directly connected to the memory modules).  As you can see from the CPU block diagram, there are items that directly talk to the CPU, like the memory, and others talk to the CPU via the Intel Mobile Chipset.  The chips that got bent just so happened to be associated with the memory of the system.
Carefully bending the pins back to being straight should fix the issue.
Going forward, while I do wholeheartedly that computers (not limited to Macs) need periodic maintenance that includes replacing the thermal paste between the CPU/GPU and heat sink components, care must be taken when doing this.  You shouldn't need to to remove the processor(s), but should it be necessary, extreme care should be taken.