I don't compete, but:
You did say "beyond the basics", but some of these things, like the portable computer, might not be obvious if you don't think about it
Ordinary clothing - indoor, outdoor, and whatever you need to stay warm at cold rink buildings
Skating outfit
Makeup & sparkles if desired or required
If you are on a Synchro team, and there is a club jacket to wear when you aren't skating, bring it too.
Your pillow and/or special blanket and/or silly stuffed animal, if needed
Deodorant
Toothbrush & toothpaste
Feminine hygiene supplies (wouldn't apply to me - I'm a guy)
Things to tie up hair, if needed (wouldn't apply to me, because there isn't enough to tie among other reasons)
Hair dryer (ditto)
Medications, vitamins, sports drinks, etc., that you need
Wallet with money, credit card
Coins, for laundry machines, vending machines, etc.
All the IDs you might need -
Driver's license, or whatever passes for photo ID in UK
Skating club membership ID.
Don't underestimate your expenses.
Checkbook
Your purse or backpack, if appropriate
A lock, in case there is a locker in which to leave things; whatever you need to unlock it (!)
Any paperwork needed to show you have paid to compete (Note: many competitions require early pre-registration).
Paperwork showing registration and/or prepayment for lodging
Required documentation for music - e.g., sources, and if required, permissions.
Check ahead with competition to find out what you need
Your skating routine. In case you get nervous. Also, some competitions require a copy of the routine
Laundry bag
Laundry detergent (even if you bring enough clothes, unexpected things can happen; if you need a super-gentle
detergent for some of your outfit, bring that too)
First aid supplies, e.g.
Paper first aid tape, which I use instead of band-aids or other adhesive strips
Cloth first aid tape to reinforce ankles if needed
Duct tape, fabric tape, needle and thread, needle threader, fabric glue, scissors, for emergency repairs of your skating outfit.
Something to read or play with (e.g., tablet) - people wait around a lot at any competition.
A portable computer (PC or Mac) so you can browse the whole Internet with for things that you didn't expect to need
Cell phone for emergencies and contacts
Chargers for cell phone, tablet, computer - I see people at the rink all the time who have forgotten chargers.
Multiple sources of light (flashlight, cell phone app, bio-luminescent sticks, whatever)
GPS - I "know" you can't get lost, but sometimes roads aren't marked.
Watch and/or alarm clock, in case cell phone fails.
Extra batteries, for everything electronic. All batteries should be fully charged.
Contact info for rink and lodging, on paper.
Contact info for your friends, so you can brag about your gold medal
Doctor, coach &emergency contacts, because some competitions require them
Pencil, pen and paper - just in case
Map & compass, in case GPS fails, with destination rink & lodging addresses, route, pre-marked
Music in MP3 jump drive (USB Flash Drive) form too - because some competitions don't take CDs
Your routine exercise equipment
Roller skates or folding scooter, to get around if needed - or extra money for a taxi
A binder or tie-closure envelope to put papers in, along with your gold medal
Yourself, confident, relaxed and happy.
I don't know what rules the UK has regarding transporting things on trains. That might change what you can bring. E.g., laundry detergent, batteries, etc. Are skates a problem?
However, you shouldn't carry so much that you get tired before the competition. Pack ahead of time, and see if you can comfortably carry everything. If it all needs to go in overhead bins, see if it is small and light enough to be practical and meet all train system rules.
Get organized ahead of time - pack separately what you need to bring to the competition building, from what can stay at your lodging