It's entirely reasonable to be worried about your child when symptoms seem concerning. Did you specifically mention brain tumours as a worry when you took your child to see doctors? Did you receive an explanation you found reassuring?
Frequently with children, time is one of the most important investigations, to see how the symptoms evolve. If the headache persists it would be entirely proper to see the GP again for a review within the next few days. This is a better route into the system than going privately for investigations for most circumstances involving children. All the services for care of children's brain tumours are within the NHS and the scans / decision making should be done by people with expertise. As has been mentioned it is not trivial to get good images in a child of this age.
HeadSmart (www.headsmart.org.uk) is an excellent resource covering symptoms of brain tumours, aimed at patients and healthcare professionals. If you think the symptoms fit anything described at your review then you can draw their attention to that specific item.
GPs are generally speaking good at managing uncertainty even though their range of experience in specific conditions might be limited - brain tumours are rare in absolute terms. If you are re-attending with the same problem, a worry about a serious diagnosis, and a sensible resources that substantiates your worries, they are likely to either:
a) acknowledge your concerns and recognise a situation requiring shared decision making.
b) feel intimidated and unable to provide sufficient reassurance on their own.
Either way you are quite likely to end up with a referral to a paediatrician if that's what is indicated in this situation (pretty urgently if there are concerning features). If you end up not being referred then you can ask for specific safety netting advice ie "what should I be looking out for, in case it is a tumour"?
If you have specific diagnoses in mind, it's always better to state these (even though that can be scary) rather than hoping that doctors will come to the same conclusion as you're reaching.
I hope that's helpful. I have tried to keep this advice generic as I'm not your doctor.
Main take home for right now is that private scans are not easily arranged, not cheap, and even if they find concerning things need the NHS based system to actually admit and treat - waiting for private things to come through rather than re-presenting when the picture changes can create more delays than it solves. I understand you feel dismissed by the doctors you have seen so far but that doesn't ever mean that a new clinician won't be looking at it with fresh eyes.