I'm fairly certain "dump" is the more commonly used word in the UK. There's the classic children's novel "Stig of the Dump" published in 1963.
There's definitely cases where UK English has adopted a new word but American English hasn't. I was surprised to see the word "coupon" used in old UK literature. "Voucher" is the more common word nowadays. People would accuse you of being Americanised if you said coupon.
Tip is definitely more common everywhere I've lived in the UK.
Stig of the Dump is set in an informal place people dump rubbish, if I remember correctly, which is where people are more likely to use the term (e.g: "what a dump"), but if you are talking about a managed place it is taken to be disposed of properly, tip is by far the more common term I hear used.
Dump and tip have different connotations. The tip is where one deposits refuse/rubbish. Whilst a dump and is a pile/collection of unwanted things, not necessarily rubbish.
It’s a small but often important distinction.
As a less confrontational reply: from my point of view “unwanted fly tipping” is talking about rubbish. The concept of depositing unwanted, yet usable, not rubbish, things somewhere is not something that would happen (outside of the rather recent disposable culture context).
As to your sentence (which is perfectly fine), the use of words avoids possibly confusing repetition. Atypical use would be preferred to “no fly tipping, take it to the tip”.
I love Stig of the Dump. I read it at school and whilst I have very memories of my childhood I do remember reading it. I can quite imagine living in a dump/tip with old jam jars as my window!
There's definitely cases where UK English has adopted a new word but American English hasn't. I was surprised to see the word "coupon" used in old UK literature. "Voucher" is the more common word nowadays. People would accuse you of being Americanised if you said coupon.