tip number 2 

If they say something like, "See you, Jimmy. Any more trouble from you and ah'm goaneh shove yer teeth so far down yer throat ye'll need te put your toothbrush up yer arse te clean them!", they're usually Scottish.

Dave (a Scottish human) 

Being Welsh, Scottish or Irish is a question of ethnicity and or/place of birth.

The equivalent in France is that a French citizen may be Breton or Basque, for example. In most cases, a Breton or Basque person is born in Brittany or the Basque country, has a strong sense of regional identity, a Breton or Basque name, and in some cases they may speak the Breton or Basque language.

In the UK, it is similar. A UK citizen may self-identify as English, Welsh, Scottish or Northern Irish, and there is usually almost no question about which one somebody is. A person who has a Welsh name, was born and brought up in Wales and either speaks the Welsh language or speaks English with a Welsh accent is definitely Welsh! There is no doubt about it. That person is British, but not English. A Northern Irishman will have a UK passport, but he is most definitely not English, either.

Obviously, it's not always quite so clear-cut. There has always been free movement between the four countries in the union. Someone may have one parent from one country and one from another - a person may have been born in England with a Scottish mother and a Welsh father, for example. In that case, they could say that they're half Scottish/half Welsh, and they'd probably be eligible to play for England, Scotland or Wales.

When it comes to eligibility to play for national teams, there are various regulations. You can usually eligible to play for a Scottish, Welsh or Irish team if you have lived in that country for a specified number of years and/or if you have a grandparent who was born there.

I think this is what you're looking for:

https://en.wikipedia.org/wiki/FIFA_eligibility_rules

"There is a specific agreement, stipulating the conditions to play for a national team, for the four British associations. Besides having British nationality, the player needs to fulfil at least one of the following conditions

a) he was born on the territory of the relevant association;

b) his biological mother or father was born on the territory of the relevant association;

c) his grandmother or grandfather was born on the territory of the relevant association.

If a player has a British passport, but no territorial relationship as provided for in conditions a-c above, he can choose for which of the British associations he wants to play."

I am not a public international lawyer, but I've thought about the issue of the disintegration of the UK for a few years now.

If the UK disintegrated, all international treaties signed by the UK would in theory have to be reviewed because the identity of one of the parties would have disappeared.  This would include treaties with Crown Dependencies like the Channel Islands.  

In practice, it is possible that the countries of the UK would agree that England, as the largest country in the UK, would assume the legal responsibilities of all international treaties; however, any such intra-UK agreement would not bind other parties. 

The case of Gibraltar may be different because the Treaty of Utrecht was signed by Great Britain, not England, in 1713.  It provided that, if Great Britain wished to relinquish Gibraltar, then Spain would have the right of first refusal. Spain may not accept England's assumption of Great Britain's responsibilities under this treaty, and argue that the treaty had become void. 

Ultimately, international law is highly political and politics is more important than law.  There is often no court which has jurisdiction over international disputes, and even if there is, enforcement depends on the prevailing political reality. 

So in short, no one can be sure what would happen.

If you're born in England, you can play for England. If you're born in Wales, you can play for Wales. As all four home nations have a national football team, it is clear cut. 

It must be borne in mind however, that a player born in England could play for another nation due to the nationality of their parents or grandparents. For example, you will find a lot of players who were born in England have played for Ireland and of course it happens internationally (e.g. Thiago Motta).