A lot of DD's class have the Huawei Ascend G300. It's on Android so they can all have the same apps and is an excellent budget smartphone. DD(11) has it on Vodafone, which has a built in Guardian app where you can restrict incoming calls from unknown numbers, internet usage etc.
If you were ok with getting a Windows phone, the Nokia Lumia 520 is the best budget smartphone at the moment on spec vs price.
DD sends texts via Whatsapp and a £5 top up lasts her for a good couple of months. She doesn't use it to make calls or browse the internet though. She's happy with playing games and texting her friends.
OP, I recently had a similar dilemma with my DS who was 11 at the time and wanted a 'good' smartphone.
What I ended up doing was buying him a used Samsung Galaxy S2. It's a good couple of years old now but in terms of screen and performance it's still very good. Certainly more than enough power for a child.
The phone cost me £160 and it lasted a total of 4 months before the muppet spilt a drink on it and broke it. I haven't replaced it as such because I needed to teach him a lesson.
You may wish to also look at the HTC Sensation. These are selling used on eBay in good condition for around £100 mark. That is also very good phone which should be enough for a child.
I've put my DS on a Tesco SIM only contract for £7.50 capped. First month he bust his data limit after two weeks as he'd downloaded games which was chewing his data. Lesson learned for him. But the cap is absolutely essential.
The bottom line is that I think it's better to get them a used phone which was top of the range a couple of years ago, than a new mid to low range phone. It will save you money, not tie you to a contract and just be less hassle all round when if they break it.
There are certainly some SIM only deals with no included data. I think you can use sliders to select how much data, how many texts, monthly spending limit, and so on (and thus which deals are shown) on e2save.com and www.mobiles.co.uk and possibly on carphonewarehouse.com (owner of e2save and the other site!!!)
Note, that only compares the main networks (Orange, T-Mobile, Vodafone etc) and won't mention rates on Asda Mobile, Tesco, Ovivo mobile, etc.