Any modern form of Internet connection (ADSL2+, VDSL, GFAST, FTTP, Cable, satellite, 4G/LTE and 5G) should do these days unless your still using ADSL, dial up/ISDN. Netflix can do 1080p at 2Mb/s due to improvements to encoding and using adaptive bitrates. I understand iPlayer can do this too. At lower speeds, you will get an inferior PQ to people watching higher bitrates, but its not the all or nothing it used to be.
The universal service obligation (USO) of broadband is 10mbps, if you don't currently get this then you should have options.
www.ispreview.co.uk
Most people don't realise relying on years old routers, will not get them the best speed or performance, there have been a number of new WIFI standards since even 10 years ago - 4 since 2008! Stop using insanely long cables (ethernet is fine - works upto 100m, but not the cable that connects to your phone socket). If your equipment is older than 4 years, it should be replaced, use any new provider/ contract renewal wisely. WEEE waste can be recycled through your local dump.
For people who still rely on copper phone lines, Some people have it on their extension wiring which is wrong. it should be at the master socket and this should be filtered. even worse is when the faceplate is filtered and they still have the router plugged into extension wiring! Turning your modem/router or both off at night is also ruining the connection as the automated systems Openreach use will think there is a fault on the line and reduce speeds in order to compensate, leaving it on will ensure their systems keep you at the best possible speed (this will take days to self-correct but maybe quicker if your provider or Openreach can reset it). if you are on legacy technologies like ADSL2+ (unto 24mb/s) ask your provider if you can be migrated to a FTTC service (comparison sites might help), this is likely to be faster and cheaper!
Ideally the router should be positioned as close to the centre of your building as possible (it radiates out like a sphere - the router is the core of the sphere), this should not be hidden behind furniture or near the kitchen. If you have a large house and the signal is poor, then properly configured repeaters (e.g. plume) should be deployed to aid coverage. Using ancient devices will slow down WIFI (e.g. 80211B/G) - turn these off if you can! Also, if you can select 5GHz (or even 6GHz) this will be faster than the standard one of 2.4GHz - yes this gives more coverage but it's also a very congested frequency limited to a small number of channels. If you do go into the settings and find a channel width option, keep this to 20/40MHz and don't be tempted to change this to 80/160 or higher, these will ruin your neighbours connections unless you happen to live alone in the middle of nowhere!
The issues might not even be with you in-home setup at all, it could be issues in the access network which Openreach (or virgin or ALTNETS) would need to be involved. Unfortunately this will mean a call to your provider to initiate such a thing, keep pressing if they insist their checks are OK.