I found the solution to this. Special thanks to patkim - creating a new Boot entry wasn't necessary but it led me down the right path to finding the solution.

Side note: First I expanded the E drive since cloning with Macrium Reflect left most of the space on my new drive unallocated, defeating the purpose. To do this I used AOMEI Partition Assistant to merge the two, which was successful but modified the recovery partition on the new drive, making it a standard partition for some reason (I don't fully understand this). To fix this I simply deleted it and cloned the recovery partition from the C drive again using Macrium Reflect.

Onto the solution - I opened the command prompt as administrator, and ran bcdedit which showed me device and osdevice were still pointing to the current C drive. I modified both using the following two commands to point to e instead:

Bcdedit /set {default} device partition=e:
Bcdedit /set {default} osdevice partition=e:

Then I restarted. At first I thought it hadn't worked because running bcdedit showed them still pointing to the c drive, but the drive letters had been automatically swapped.

To test that everything was working from the new drive only I physically disconnected the old one and it worked.