Europeans are advised to get a free European Health Insurance Card (EHIC) before travelling to Lapland or Iceland. This does not replace health or travel insurance, but entitles you to any necessary state-provided medical treatment.
If travelling to Alaska, you will need travel insurance that covers the USA. There is usually an additional supplement for this as US healthcare is private and very expensive.
Medical facilities in these regions are generally excellent, and each town usually has at least one 24-hour pharmacy.
Mosquitoes in Lapland in late June-July are vicious. Keep little ones well covered if travelling during this time.