Every pool is different, you can tour them in advance or just ask at desk/on phone.
1. Is it uncommon for gyms in the UK to have gender-specific changing rooms?
What do you expect families to do, dads with daughters or mums with sons, otherwise?
2. How can I find out if the gym showers are partitioned before committing to a membership?
Phone them? Ask for a tour, they all give tours before you pay for membership. And ask then. Ask them if you bring own lock or need a pound coin.
3. In shared changing rooms, how does one go about showering? Do you go in fully clothed, shower, and then manage to dress again while still in the shower area?
Is nude showering the norm where you're from?
Most places have some private cubicles for showering. You'd be expected to exit the cubicle not nude.
4. Do I need to bring my own shampoo, body wash, and shower towel?
Where are you from that all that would be provided as part of membership or swim session fee. I've managed to attend public pools in France, USA, Iceland, UK (obs)... so none of them.
5. I remember that using lockers was a bit different, but I can't recall the specifics as I was quite shocked last time. Could you explain what to expect?
In most countries you bring your own lock (definitely USA). In UK there are lots of pound-coin operated lockers (you get pound back when you open up). Sometimes you can pay a deposit for a key, return key & get your deposit back. Many people in many places bring their bag of stuff somewhere poolside so they can keep half an eye on it. This was officially banned in France but lots of people did it (typical French)
6. Are there any other etiquette norms or things I should be aware of?
Don't impose your nudity on others. Brits are quite prudish about it. Some pools have rules about must wear caps or flipflops or shower 1st (not that everyone observes all such rules). They will tell you when you join.