Look for a Podiatrist with HCPC (Health & Care Professions Council ) it covers the standards on ongoing CPD (Continual Professional Development )
Be aware in Private Practice there is no legal requirement to be registered , it is not a closed profession
Or if you get an NHS ref via your GP you will have an HCP registered practioner its a condition of employment
In the meantimes DON'T poke anything down the side of her nail , you risk introducing injury , portal for infection and injury to the nail , not to mention your poor daughter being scared and hurt .
Boil water , add salt (enough to go cloudy) , let it go tepid then bathe the toe 5-10 minutes .
Air dry and dry sterile gauze dressing
No creams . The salt will dry it , draw any fluid , its a natural antiseptic
Keep a check on any pus that comes out . Any redness (signs of infection)
See your GP /phone them if theres any sign of infection (might need antibiotics)
When you see someone they should be able to remove the spike and smooth the edge with sterile instruments , then advice on dressings etc and if she needs removal
Injections are not as bad as you think but personally , I wouldn't inject for the initial treatment , I'd want to know the toe felt better , which she won't feel if its numb