If there are more than 10,000 people in a British town, can they get a uniquely beginning postcode?

Post codes in the UK are extremely complex.
They are normally in the form LLNN NLL
Where L - Letter N- Number
The first LL is the Postal town then NN is office reference NLL is the street reference.

This explains it better: Postal code - Wikipedia, the free encyclopedia

The first 1 or 2 letters of every postcode in a town are the code for that town.

For example BS for Bristol, CF for Cardiff or G for Glasgow. So I'm not sure what you mean by a "uniquely beginning" postcode, as everyone in a town will have the same beginning of postcode, so it is not unique.

Could you elaborate on your question a little more?