Here is a quick reference that shows difference between various chroma formats - 4:2:0, 4:2:2 etc. As can be seen that in both cases, chroma samples are not aligned with the luma samples. Hence, for proper pixels reproduction filtering is required before the display can be applied.
Probably one of the best source to look at is the MSSG (MPEG software simulation group) which is an official reference code base for MPEG 1 & 2 encoding and decoding. If you look at the code, you will realize the conversion from the 4:2:0 to 4:4:4 is done in two steps i.e. 4:2:0 to 4:2:2 and then 4:2:2 to 4:4:4 (horizontal). It uses 5 stage poly phase FIR filter which you can sneak into the code.
However, the standard doesn't really specify which filter should be applied.
Here is one of the best reference that describes the effect of different filtering and provides further references as well.
In practice, chroma subsampling artifacts for 4:2:2 and progressive
4:2:0 formats are rarely noticed even where it is poorly implemented
(e.g. with nearest neighbour or box resampling). In particular, 4:2:2
is commonly referred to (and sometimes marketed) as “visually
lossless”, even though it is not actually visually lossless in all
circumstances (e.g. red text on a black background) . But while
chroma subsampling is not entirely visually lossless, it seems to be
good enough that many people do not notice otherwise.
On the other hand, 4:1:1 and interlaced 4:2:0 formats can be
problematic as they effectively subsample the chroma by 4X in one
direction (interlaced 4:2:0 effectively subsamples 4X vertically since
each interlaced field is subsampled individually).