This is not commonplace. Most operators in the UK get along fine with the 400W limit. Given the path losses inherent in Earth-Moon-Earth - hundreds of decibels - the antenna is a much more important factor than is the power. You need every extra bit of push you can get, but if you're not making it at 400W, 1500W isn't likely to be much better.
Remember that doubling the power equates to only half an S-unit at the receiver. Even going from 400W to 1600W would only give you an extra 6dB gain - and wouldn't help you receive other people's signals at all. Invest in a better antenna rather than a better amplifier, you'll see much better reception in both directions.
The process you'd have to use is an application for a Notice of Variation. These are issued by Ofcom, and there is an opportunity to apply for one for Special Research - but you will not get it granted solely for the purposes of higher power for EME. This is the process used for example by the hams that are experimenting with lower frequency communications, like 500kHz and 2200m, before they are officially authorized to all operators.