What do babies wear in nursery when it's colder? Help!

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Our 13mo will be going to nursery soon and I'm thinking through what clothes to send her in. Nursery have said waterproof and coat and warmer clothes when needed. In nursery they go on a few trips in the big buggies and play in the garden.

When it was last cold she was obviously a lot smaller and she was in pram suits in pram with a foot muff.

What I'm struggling with is firstly she can't independently stand/walk yet and does a lot of crawling around or pulling to stand with support, so any warm clothes would need to keep legs warm and dry too. And she doesn't have shoes yet as doesn't need but perhaps I buy now?

She's also quite skinny, 8.4kg ish in weight (9th percentile) so trousers don't stay around her waist, she wears tighter leggings but they're obviously quite thin and not suitable for winter.

Any help from those been through this please? Hoping to buy a few bits this weekend so it's sorted. Thanks!

Given they were also be inside for some of the time I sent mine in leggings or trousers and a vest and t shirt and a hoody/cardigan/jumper etc. They then had a fleece lined puddle suit or waterproof dungarees and a coat for when they went outside. Essentially they need layers so they can be comfortable both inside and outside. They went in shoes and took wellies too.

You can buy lined leggings if you want some warmed ones for winter.

The rooms themselves will be warm so dress normally.
For outside I second a fleece lined puddle suit. The thicker outer ones last a long time and are durable (think fisherman coat fabric not thin cagoule).
Baby bogs are great and have good resale value.
Mittens on strings, all of the cheap hats and we had a weird polo neck scarf thing from H&M it went over the head and looked like a knitted polo neck with a square on the front and back to keep chest and back warm.

Fleece lined puddle suits from M&S. Waterproof booties to wear over wee soft sole shoes.

Dungarees, long sleeve t shirt, cardigan or zip up hoodie and puddle suit works well for skinny ones, once walking they will need suitable footwear for outside too

As a nursery practitioner in a 0-2s room, I would advise against very warm or fleece lined indoor clothes, nursery rooms are extremely warm and she’ll overheat. For outside we ask for wellies, a warm, lined puddle suit and a hat. For inside, just normal clothes, leggings, joggers etc, with T-shirt and jumper or cardigan and slippers. Plenty of layers and spares (including socks!) in her bag, we go through a lot and most parents underestimate how many will be needed, especially if their child enjoys messy play or getting muddy in the garden.

Also, unless dungarees have poppers underneath for nappy changes, please avoid them. Changing 12 plus nappies takes ages and having to strip multiple children down to be able to change them delays everything else and takes a staff member off the floor and essentially out of ratio for longer.

DD has skinny legs too so I’d put tights under leggings/dungarees etc to keep her extra warm/pad legs out. H&M and a few places do fleece lined leggings with adjustable waists. Thermal socks (home bargains do cheap ones), welly socks, fleece lined puddle suit, waterproof fleece lined hat (mountain warehouse/ Next), slippers or lined crocs for indoors