UK has the state pension which comes from NI (National Insurance) contributions which in a way acts like a Defined Benefit pension in that you work for X number of years and get state pension of Y in return(currently adjusted for inflation, wage growth, or 2.5% annually, called the 'triple lock'). Not based on income so you don't get a massive state pension by earning 6+ figures.

Then more recently (as in, as of around 2012 and up to 2018) we got auto-enrollment into private pensions, which are more like Defined Contribution (DC). Employer has to pay a percentage into the pot and so do you. Usually 5% employee and 3% employer by default but many will offer better (or contribution matching) as a perk. I think this is probably the same or similar to the 401k in US terms. The employer chooses the pension provider but you need to proactively switch to a high risk scheme to see any growth from it.

At a certain point the tax rebate from the government doesn't cover your whole income so you have to file a tax return to get the rest of the rebate. You can instead choose to 'salary sacrifice' which means you are lowering your income on paper but the sacrifice is put directly into pension (or otherwise can be used to get a car on lease or a bicycle via another scheme). Salary sacrifice is used by a lot of higher earners to bring their gross income down in order to avoid being cut off on certain benefits like child-care.

After all that you have SIPP (Self-Invested Personal Pension) which gives you more control over what you can invest it. Not just stocks, ETFs, and all that, but can also be commercial property (so the pension itself owns that asset). This gets the same tax treatment for pensions.

Finally there is the ISA and LISA. The first is a savings account where any interest or capital gain is free of tax, the second gets a 25% boost by the government to help buy a house or flat, but you can only use that money for a mortgage deposit.

Most people won't see all that much from their auto-enrollment given they could just opt-out to get the extra cash in their paycheque (especially when a low earner), or might not know to switch to the high risk fund, so the state pension and other benefits for OAPs will be there still. Those with more disposable income or a long term view (e.g. doing FIRE) are likely to max out the various vehicles available to them but at that point you're gonna be earning too much to care.

No need to optimise healthcare costs or any of that unless you want to go private.