This response relates only to the Qualified Tuition Program, or 529 plan, part of taxi’s questions.
The legal basis of a QTP is not completely clear to me. The best interpretation I can make is that it involves a gift to the relevant child, and that the fund is the property of the child, albeit with restrictions on access to it. It does not appear to involve a trust. Perhaps others would comment on this.
Some or all of the income from the fund is sheltered from US tax. However, this tax exemption is not mirrored in the UK tax system where general principles apply.
The income which arises from the QTP is taxable on the child (and not the parents or grandparents). The UK has a tax rule which provides that the income which arises to a child under 18 which derives from a parental gift is taxed on the parent. Here the gift is from a grandparent and the child is also presumably over 18.
The child has a normal UK personal allowance, £10,000 for the current year, as this applies to all UK residents.
The double tax treaty does not affect the situation. The treaty does not apply to US citizens other than some exceptions. One of these exceptions is Article 20 (Students). This article, which has various conditions, only exempts income for the maintenance education or training of a student. This could, for example, exempt from UK tax a US scholarship. However, it is difficult to see how it could exempt income from a QTP. Again, others may care to comment on this point.
Gains from sales of investments would also be taxable on the child (on the above interpretation of the legal basis of a QTP). The capital gains tax annual exemption is currently £11,000. If, however, gains arise from the sale of non-reporting mutual funds, they would be taxed in the UK as income under the offshore income gain rules.
The remittance basis is not likely to have any application to this situation.
If the income of the child (from all sources) is below the personal allowance and any gains (again from all sources) is below the annual exemption, then there is no obligation to report anything to HMRC or to complete a self-assessment.