These accounts are through a large multinational company based in the US.
Although they may be linked to others, financial institutions have to exist in one jurisdiction, because regulations vary. (TTBOMK even within the EU, though less so.) If you have an account with a US company, including a US subsidiary of a non-US company, that's one thing, and if it is a non-US company, including a non-US subsidiary of a US company, that's different.
Assuming these accounts have mid-to-high five figures USD total, will we need to pay taxes on the money in these accounts?
US Federal tax does not apply to assets regardless of location. (Except at death, and then only for the richest few.) Most (maybe all) US localities have property tax on real estate, and sometimes some other assets, but not foreign ones.
For a US-resident alien (or citizen), there are requirements to report some foreign assets, under two separate but overlapping schemes -- and if you are required to report and don't there are penalties that may function like a tax but aren't called a tax.
for 'bank' accounts, which is actually defined to include some other things, if the total value is over $10k you must report on FBAR to FinCEN, not IRS. This may apply to her depending on details of the accounts.
for 'financial' accounts, a slightly broader category, if total over a threshold which for your case (married filing jointly, resident in US) is $100k at end of year or $150k anytime during year, you must report on form 8938 in your tax return. From your description this should not apply to her-and-you.
The IRS, although it doesn't handle FBAR filing, does have this nice comparison chart.
Income (earnings/gains) on these foreign assets, like that on US assets, is generally taxable no matter where it is from, once she is a US tax resident.
Investments for retirement
If these are private investments, their puprpose (for retirement) does not affect their tax treatment. Dividends or distributions received, and realized capital gains net of losses, are taxable, although the rates vary and if your income is low enough some may be taxed at 0%; see my answer here (except that most brackets and standard deduction are doubled for a joint return, and all of them adjust for inflation in future years). Unrealized gains are not taxed. (Possible but unlikely exception: as a 'permanent' resident aka green-card holder if she leaves the US and abandons that status or it is revoked, and has levels of assets or yearly tax substantially higher than your Q implies is currently the case, she could be subject to the 'expatriation' tax on unrealized gains.)
Added: as commented by eric if these investments are in non-US companies, they may be subject to the 'PFIC' (Passive Foreign Investment Company) rules, which are more complicated and sometimes require you to pay tax on undistributed income or unrealized gains. (Your Q is ambiguous because accounts with a possibly-US company could still invest in non-US things, and vice versa.) As the name says, PFIC is aimed to cover mutual funds and investment trusts, but the way it is defined can sometimes accidentally pull in 'real' (operating) companies. I, as a US native, have occasionally invested in non-US companies and their prospectus and annual reports contain a section on 'US tax consequences' that says something like "we/$company believe we will not be classified as a PFIC for US tax purposes, but this cannot be determined until the end of each tax year and there is no guarantee it will not happen; if it does, US holders will be subject to additional tax". Handling this case by hand can be pretty complicated and I would suggest using software, or possibly a preparer.
Government-run retirement schemes that are considered 'similar' to US Social Security are not subject to US tax, and I'd expect Singapore's CPF falls in that group, though I don't know where to look for confirmation.
A life insurance policy
Earnings within a 'whole' life or similar insurance policy are not taxable. except if you receive distributions exceeding the premiums you have paid.
A bond which will mature / pay out in her 40s, about 10 years from now.
Interest on a bond is taxable (unless issued by a US state or local government agency, which I presume this is not). If the bond was issued or she (subsequently) bought it at a more-than-de-minimis discount from face value, then a portion of the discount is treated for tax purposes as received each year (amortized) even though it isn't actually paid until maturity (or earlier call/redemption or sale, where permitted); this is called 'original issue discount' (OID) or 'market discount' as applicable. (The amount taxed each year is added to her basis, so that when the payment is actually received there is no or little reportable gain for tax purposes.) The computations for this are pretty complicated to do yourself, but see below.
Are there any tax forms that the company she is investing with needs to send us? (And will they be able to send it?)
If this is a US financial firm, she should have previously been solicited for a W-8 series form, probably W-8BEN, to establish she was -- then -- a US non-resident
and potentially claiming any benefit as a resident of a country that has a tax treaty with the US (according to the current list Australia does, Singapore doesn't).
At least for the 'investment' and 'bond' accounts; for life insurance probably not, since as noted it usually is not taxable. Assuming so, she should inform the company her status has changed, and probably provide them a W-9 instead of the W-8. After that, they should provide her, and also file with IRS, any required 1099 forms. In particular if bond discount applies they should handle the amortization compuation and provide it on 1099-OID. Although the 1099's and 1042's for 2019 may already have been issued, or at least prepared, and they may or may not be able to correct them this late.
If they don't, or if this is a non-US firm and thus not subject to US reporting requirements (other than the new FATCA ones, which work differently), you are still required to report everything correctly. If you have the bond discount situation and don't get 1099-OID for it, you could look at the instructions in pub 550 (note 2019 hasn't been released/posted yet, and based on recent experience it may take until March or even April, although I don't think there will be material changes in this area), but personally I would suggest you get help -- either from an actual preparer, like a CPA, or by using software for your return if it handles this. (I don't know because the only time I had bond discount was back when I filed by hand on paper, and I did get 1099-OID from my broker.)
Good luck :)