The first question is if you're saving (cash - with interest, guarantee no loss of principal) or investing (stocks & shares - higher long-term expected returns, but capital is at risk, will be more volatile).
If saving, the main options are:
1. Savings account: either in your name, or in the child's name. Some of the child savings accounts have pretty good interest rates. Interest will be US and UK taxable, but Personal Savings Allowance applies for the UK, US has some thresholds below which child interest doesn't have to be filed ($1,500 this year).
2. Premium bonds: prizes not interest, but with a decent holding and a long enough time period, statistically very similar. UK tax free, US taxable. You can hold up to £50k in your name, £50k for each child.
3. Cash ISA in your name: basically the same as a savings account, except UK tax free.
4. Junior cash ISA in the child's name: same as a cash ISA in your name, except that it becomes the property of the child when they turn 18. Your call if you expect your kid to be responsible enough to handle it. Key constraint is that you can't take any money out until they turn 18 (or become terminally ill or die - hopefully not).
5. US savings bonds (I or EE): if you can be bothered to deal with the archaic TreasuryDirect website, these are potentially interesting options. US and UK taxable, although you can decide whether to pay the tax every year or wait until maturity.
Investing Options
1. Junior S&S ISA in the child's name: you'd have to do individual stocks, unless you can find a way around the PFIC/KID catch-22 (although the KID part is in consultation to repeal, which would make it easy to invest in US ETFs to avoid the PFIC part - make your voice heard:
https://www.gov.uk/government/consultations/priips-and-uk-retail-disclosure). Same as the cash junior ISA in that it becomes the child's property at 18, and you can't withdraw until then. US taxable and reporting on individual stocks is a bit of a pain (doable, just tedious).
2. Adult S&S ISA in your name: same PFIC/KID catch-22 as a junior S&S ISA and US taxable. Key advantage is that you can access the money any time, not only when they turn 18, and it's your money - you can gift it at 18 if you want to, but you could decide your kid isn't ready and wait.
3. Taxable brokerage account, probably in your name. Have to navigate the PFIC/KID catch-22, but if it's a US brokerage who doesn't know/doesn't care that you're a UK resident, you may be able to buy US index funds (you'd want them to be HMRC reporting:
https://www.bogleheads.org/wiki/US_domiciled_ETFs_that_are_UK_HMRC_reporting_funds)
4. UK SIPP/pension: either your own pension/SIPP, and then you gift to the child later (if the timings line up), or you can open a child SIPP. This is a
really long term investment, they can't access until age 58ish (likely to go up over time). But US and UK tax advantaged, with all the usual grey area caveats around a SIPP (foreign grantor trust, form 3520/3520A, etc.).
5. IRA: If your son had earned income, he could contribute to a Roth IRA. Probably not at 6 months old
But could consider if/when he gets a job. IRS doesn't care exactly where the money comes from, so entirely fine for him to keep his money from the job and you to contribute to the IRA, up to the amount of earned income (or the annual max).
Frankly, none of the options are perfect, but depending on your priorities, your timeline and your appetite for managing individual stocks, there are options to consider.
Personally, I've got premium bonds in my kids' names. Everything else is in my name, and if/when the time comes, I'll gift to them. I don't expect to be anywhere near US gift tax lifetime limits, and for the UK I just need to not die within 6 years of the gift (other options are available, like annual limits, but that's the very simple one). Also gives me control - if I don't think they're ready for the money, they don't get it. Equally, if there's a good reason even before they turn 18, I can access the money, at least from the accounts without restrictions on my age.