This reminds me of Betjeman’s poem ‘How to Get on in Society’ (see below), which is basically a satirical attack on Britain’s aspiring middle classes. Notice the first line of the last stanza ‘Milk and then just as it comes, dear?’.
UDS is correct, the true upper classes would never think to put their milk in first and would all pejoratively refer to people who do as a ‘bit milk in first’. This is similar to the HKLP syndrome (“she’s / he’s quite HKLP”), as in ‘Holds Knife Like Pen’.
It is my understanding that these are all uniquely British characteristics/prejudices, which came about during the expansion of the middle classes during the 19th century. Things that seemed to be the ‘correct’ way to do things were quickly adopted by people who wanted to elevate themselves from their lower/middle class backgrounds. An example of this is the HKLP issue: this could easily be perceived as the more ‘correct’ way of holding your knife. In reality, the HKLP people only served to distinguish themselves from the crowd by becoming either a source of laughter or perplexion for almost everyone that doesn’t HKLP (upper/lower/whatever class).
The same is true for Milk in First: it may seem that expensive China would necessitate a cooler temperature and hence the reason for putting the milk in first. However, it is of course a completely ludicrous idea that the upper classes would only have pointlessly delicate china cups.
In Betjeman’s poem below, he also mockingly portrays the use of French words, such as ‘serviette’ instead of the more traditional ‘napkin’ (similar to ‘toilet’ instead of ‘loo/lavatory’ etc). Speaking French was something that the bourgeouis aspired to and hence the infusion of French words into the English language. However, such words are (rightly or wrongly) totally scorned by the UK’s upper classes.
Am sure that this will amuse the OP. How could even the most mundane issue highlight such absurd class distinctions? It’s a funny country generally though. You just get used to it. 
How to Get on in Society
Phone for the fish knives, Norman
As cook is a little unnerved;
You kiddies have crumpled the serviettes
And I must have things daintily served.
Are the requisites all in the toilet?
The frills round the cutlets can wait
Till the girl has replenished the cruets
And switched on the logs in the grate.
It’s ever so close in the lounge dear,
But the vestibule’s comfy for tea
And Howard is riding on horseback
So do come and take some with me
Now here is a fork for your pastries
And do use the couch for your feet;
I know that I wanted to ask you-
Is trifle sufficient for sweet?
Milk and then just as it comes dear?
I’m afraid the preserve’s full of stones;
Beg pardon, I’m soiling the doileys
With afternoon tea-cakes and scones.
John Betjeman