Yep, it's a doddle. My instructions make it look harder than it is!
Getting the back off is the (relatively!) physically tough but simple bit, I used a hairdryer on it for a good 2-3 mins all around before using a plectrum/lollipop stick to prise the back off starting at the speaker hole at the top end.
Once the back is off try and clean up all adhesive whilst it is still warm, from the back and off the frame.
Don't lose the soft foam that acts as a compress for the battery terminals at the bottom.
On my Z1 compact I had to remove the Bluetooth aerial (a white square sticker) from the battery itself, take extra care here and look how the contacts are supposed to sit for when you reconnect.
Using something more gentle than the small screwdriver I used, pull up the connector block from the battery to the phone power input, very gentle pressure required. It's at the bottom left.
The battery is held in by a single small Phillips head screw at the top, once that is removed the battery folds out from the top and unhooks from the bottom.
Put the new battery in the reverse way paying attention to where the screw and contacts go (I think it's one way only fit), and screw back in using the original screw. Gently press the connector block back in on the bottom left.
Re-attach the Bluetooth sticker CAREFULLY. Obviously if the contacts do not line up it won't transmit signal. I planned to use tape to ensure it stuck securely but actually it was fine without.
But the foam compressor back in above the power block (I guess to stop it accidentally pinging out if dropped) and prepare to reattach the back. Don't lose the lens cover from your back.
Usual stuff: clean, dry no grease, if possible in a dust free area. Common sense really I suppose, line up as carefully as you can with the new adhesive already attached. get one corner sorted and line up, then commit fully and press the back down. You probably have 1-2 seconds to shift it around a bit to make sure it fit's in the recessed frame before it becomes unmovable.
Regardless of whether you think you are successful or not put the phone face down on a covered table, and pile some books on into compress the back down. I left mine twenty minutes.
Bluetooth should still work if the contacts were lined up fine, and the edges should still feel smooth if applied correctly. I haven't tested if it is still waterproof but I see no reason why it wouldn't be!
Good luck.