Hi and welcome Emm
It must be incredibly tough for our supporters, even more so when the person with CH is a child with CH, plus migraines and tension headaches.
Are you working with a headache specialist? GPs and even most neurologists just don't have the skills and experience to deal with CH, never mind the combination of multiple headache types too. Since you only mention a pediatrician, it is highly unlikely they have the required skill and experience for anything beyond migraine, which is why they are probably just treating it as just migraine.
Do you know about OUCH UK (Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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This is a UK based support group who have plenty of experience with dealing with the NHS. They also have a support phone line and forums. There are also people here from the UK, so they may be able to give some UK centric ideas and advice.
Whilst most people with CH just have CH, there are quite a few of us who have multiple headache types (including me), with CH and migraine being probably the more common combination. This certainly makes it harder to treat as whilst many of the CH medication is also migraine medication, there can be times when treating one limits the choice of treating the other.
For both CH and migraine, you can take preventives which can cut down how many CHs / migraines she will get. The challenge is to get something which works for both which work well for her, especially being a child too.
There are also abortives, to kill off a CH / migraine. Many will work with both, although for CH the preference is to go for faster acting abortives. Again it is a complication in that she is a child, so working with the headache specialist is essential.
After a CH, especially one which hasn't been abortive, it isn't too unusual for the person to feel worn out, especially when it has been a severe and long lasting one. And with migraines, going to sleep is a pretty effective way to deal with it (although going to sleep with a CH just isn't an option).
There are multiple triggers for headaches. It is possible that some of the triggers for her will be the same for each headache type and some will be different for each. Trying to identify them may be easier with a headache diary where you note when (day / time) the headache is, plus symptoms, duration, treatments (e.g. pain killer, sumatriptan) and any comments, e.g. happened just after<something>. Over time you may be able to spot patterns.
Keep reading here, you'll learn a lot and ask any questions you have. Keep us updated with how she is doing.