Good observation. It's often misquoted that the 1,000 feet is the [international] standard. The standards and recommendations for runway markings can be found in ICAO Annex 14. In particular section 5.2.6 for the touchdown zone marking (diagram shown below).
When a country decides it wants to deviate, it lists the differences in its AIP. In this case indeed the US AIP lists a difference from 188.8.131.52:
The U.S. standard places the aiming point marking 306 meters from the threshold where it replaces one of the pair of three stripe threshold markings. The 306 meters location is used regardless of runway length.
(Note: 306 m is ~1,000 feet.)
The UK AIP does not list a difference in distance, but the reason why it's >400 m is in a note (missing from the 2016 edition of Annex 14) in the 2017 EASA guidance material, the 400 m is the standard, except:
Where a PAPI system is provided for the runway, the beginning of the marking should be coincident with the visual approach slope origin.
The AP marker has always been (using Google Earth historic view) where the PAPI lights are at EGLL. Example for EGLL 27R:
When VASI is used instead of PAPI, like the 3-row VASI at OERK, it's 400 m on the mark:
I initially thought maybe it's because EGLL went through various lengths in its long history, but a new runway such as 05R at HECA that was built from scratch, is actually like EGLL 27R. There's more to PAPI than meets the eye.
So there is diversification indeed. As for 'landing short in the US', we are looking at ~100 m difference (306 vs. 400) for the aiming point, but don't forget the extra touchdown zone marking closer to the threshold in the ICAO standard.
Note: the figures apply to runways that are >2,400 m. Shorter runways have the aiming point brought closer to the threshold.
Trivia: Also still in the UK AIP is the unique aiming point, even though they are clearly no longer used at EGLL and EGCC for example (check historic view in Google Earth or current charts).