I dont think its that different from a state school. This is the current list from the bog standard state school mine attended
Winter Uniform
School logo V-neck jumper * or plain navy knitted jumper / cardigan
Plain white shirt (not polo)
School tie (elasticated ties are available for Year R /1; children are encouraged to wear a regular tie from Year 2)
Long grey ‘school’ trousers / grey skirt / pinafore
Grey, navy or white socks / grey or navy tights
School coat or plain dark coat (waterproof)
Black or navy school shoes (not trainer style or trainers)
School logo woolly hat (optional)
Summer Uniform
Navy/white checked shirt * or plain white short-sleeved shirt (not polo)
Grey ‘school’ shorts / grey skirt
Navy blue and white gingham summer dress
Grey, navy or white socks
School logo V-neck jumper *or plain navy knitted jumper / cardigan
School coat or plain dark coat (waterproof)
Black or navy school shoes
School sunhat or plain navy, white or black sunhat
PE Kit
School PE T-shirt: Navy with red shoulder fleck *
Navy shorts or Navy skorts
Plain navy socks for EYFS and KS1; red and blue hooped socks are suggested for outdoor winter sports for KS2 *
Navy tracksuit bottoms or plain navy leggings
Hooded pullover with logo *
Trainers for outdoor use
School Bags
All children require a suitable navy waterproof rucksack or bag large enough to take A4 folders and a range of books and equipment. Bags should not carry overt brand logos such as football teams etc.
Book bag with school logo - compulsory for years R, 1 and 2 *
PE bag (with or without school logo)
Waterproof coat and wellies (Year R)
Hi-vis jacket with school logo - compulsory for all years *
- is only from school shop and they don't.mention swimming kit but do swimming from year 3 and has a similar amount to your list.
Only thing I noticed is a hat is not compulsory.