Avidskier -- my dd1 was tested at age 9 by a private Ed Psych because we had issues with the school (long & dull story) and I needed reassurance that I wasn't going mad.
She scored somewhere above the 99.9th centile, ie. in the top 0.1% of the population. At the time she was in a rough-ish primary school in a tough area, which had no interest in G&T generally, and in her in particular. She was cross with the school quite a bit, and did get pee'd off with it all, as did we. The head had taken to crossing the road to avoid us, she was in a notoriously difficult class, and had a selection of deeply unimpressive teachers.
However, however... she had a close group of friends, which stood her in good stead, and still does. She learnt that the world does not revolve around her, and that lots of people don't value the things that interest her. She had a chance to be one of the alpha kids clique by virtue of being bright and middle class, rather than becoming the class nerd, which would have been the obvious role for her to take on in a more socially and academically homogenous environment. She learnt to work on her weaknesses (social interaction) rather than playing to her strengths (academic work, obv). She acquired a good set of coping skills for dealing with being called a geek and a nerd, and a robust ability to defend her own interests in the face of scorn from her peers and lack of interest from her teachers. She gained empathy for people who learn less quickly, and realised that not being able to read at 9 does not make you a lesser species of human being.
At 11 she got into a selective state school (and yes, she scored the highest mark in the entrance exam). She has learnt that if she makes a reasonable amount of effort, she can do really well. She also does some competitive sport (well but not outstandingly so) and has a Saturday job in a charity shop. She goes camping with friends when she can, and goes to a lot of gigs. She's doing GCSE's and is aiming for 10 A*s, gawd help us, and wants to go to Oxford. Maybe she will, or else she'll go to another university and get a good degree in a subject that interests her. Coupled with her robust life skills and good sense of who she is and where she's going, she has every chance of being able to find a path in life that suits her and fulfils her. She's calm, confident, funny and interesting to be around, has lots of friends and is emotionally resilient. All these things are more important than score on an IQ test.
I have a younger dd who is cut from the same cloth as the older one, to all appearances. I have not had her tested, nor would I do so. She enjoys school. The fact that she can read the Moomintroll books at age 5 while others are on Biff & Chip Stage 2 hasn't struck her as important, and nor should it. She enjoys school (different primary from dd1), and has never complained of being bored. If she were at a pushy private school, she would probably be 6 months ahead, and would definitely have neater handwriting. But so what? Outstanding school results in Y1 are not a good predictor of a happy and successful life.
You need to decide what is important to you. If you are going to spend the next 6 years fretting that your dd is unchallenged and is falling behind in the great race for an Oxbridge first, then go ahead and scrape together the cash to go private. If she's complaining consistently about hating school being unhappy, being bored then you also need to take action (though going private may not be a panacea here). But if she is muddling along happily, doing stuff at school she enjoys, has good friends and you get on reasonably well with the school staff, then I would take a giant chill pill (set up an IV if necessary) and leave her to do what 5yo children do best explore the world, learn new stuff, have fun and make friends. Keep a gentle eye on what she's doing at school, maybe do extra activities with her, get her to learn an instrument or play maths games with you. Make sure she has activities that require her to make an effort, so she doesn't assume that she has a god-given right to be able to do everything perfectly first time. Non-academic things may be just the ticket here swimming lessons were my dd1's nemesis, and it did her a world of good to have to cope with everybody else learning it faster than she could. If round about Y4 you still feel she's being under-demanded academically, then get a good tutor for an hour a week to give her a taste of what making an effort feels like, as well as sharpening up her writing and maths skills.
At 11+ there's a whole 'nother world of opportunity out there, and by then you will have a much better idea of who she is and what she wants from life, and more to the point, so will she. In the meantime, don't wish away her primary years by constantly looking over your shoulder to see whether there isn't something better just round the corner.