Most British websites work like this website suggests, in that they ask for the postcode first, then give you a dropdown of all the addresses at that postcode.
It works great, except my address is for some reason not in whatever databases these websites use. The building number is on the list, but not the individual flats. So I have to put in the postcode, choose something like "My address is not listed", then fill in the form manually. A few times it wasn't even offered as an option.
It sounds like your address isn't listed correctly in the UK Postcode Address File (PAF).[0] There's a form you can fill in requesting a correction.[1]
For instance, my postcode covers 15 houses and half of a large park. Those houses have been subdivided into 4-6 flats, and many have been redivided or renumbered multiple times. The park also contains various buildings and other places that might plausibly receive deliveries, some of which have multiple entries (an electricity substation appears 8 times, for some reason). So in total, it covers more than 200 addresses - mine is no.140-ish in a typical sorted list.
You'd be amazed by how many address checkers can't handle more than 64, 100, or 128 addresses in a postcode. Or how many scrollbars stop working, requiring you to use the arrow keys to navigate. Or how many other glitches I've seen.
The other common problems are usually down to temporary postcodes (which used to always end with an 'X' but can now be indistinguishable from 'real' codes) escaping into the wild, re-addressings and re-numberings not being picked up properly, the old problem of outdated PAFs being used, and - my personal favourite - BT/Openreach using their own separate postcode database, dating from the Post Office split in the early 1980s, which doesn't always agree with the PAF. Agh!
Somebody else mentioned the PAF. I should further clarify why your address might (might, this could just be a mistake) not be on there and why therefore almost any form should have an option to "do it the hard way"
The PAF is about what the Royal Mail calls "Delivery Points" which are places they promise to deliver physical letters to. Inside my building for example every home has a front door, with a letter box, and letters literally get posted into my flat, but down the street there's a building with a rack of boxes set into the wall and post is delivered to your numbered box, and up the street there's a townhouse converted and all the mail just goes in one piles for everybody in that building.
Many delivery points share a postcode, but because they know them all they actually all have unique numbers, and unlike a postcode they're not for humans so they get changed quite frequently as new buildings are constructed or working patterns change, if you examine your post carefully (in the UK†) there are two rows of fluorescent orange dots on the outside, these were printed by mail sorting machines shortly after the mail was received, one is an entirely arbitrary serial number and it designates that piece of mail for a very short period (say a few days) to enable statistical tracking for performance - if #213940202 entered the system in Glasgow on Monday, was in Portsmouth by Tuesday but wasn't delivered to someone's door until Friday the problem ain't in Glasgow. But the other one we're interested in here is the Delivery Point as a numeric code. If you don't have multiple addresses this row of orange dots will be identical on every item you've received for some time yet it's different on someone else's mail.
† This trick was invented in Britain but is used (licensed) in some other countries because it's a good idea, however exactly what is encoded and so what it "means" may vary.
I had a similar problem. The way it was explained to me was that in practice, there are two databases used in the UK for this commercially and they don’t always agree. I used to live in a bungalow next to a large house that had been converted into flats. In one database it was listed with the bungalow name then the street name, and in the other database it was listed as “Flat 8” using the other building’s street address. About half of British delivery forms used one place, about half used the other.
It works great, except my address is for some reason not in whatever databases these websites use. The building number is on the list, but not the individual flats. So I have to put in the postcode, choose something like "My address is not listed", then fill in the form manually. A few times it wasn't even offered as an option.