By Prompt dates, LME means dates on which contract trades. Normally, you see monthly contracts. Prompt month is also front month. LME is providing daily contracts up to 3 months out, then weekly and so on as in your reference.
Generally prompt month is the front month or nearby month that is closest unexpired month. You can replace month by day or week as is the frequency. Probably better to talk interms of prompt contract. Trading systems have mechanism of determining prompt month ( aka front month, nearby month). By the same token for daily contracts they can have first nearby contract. you get the picture.
There is no trading without knowing the market in which you are trading.
If you are in Asia trading nn LME you have to follow LME rules. It does not matter where you are - what matters is where you are trading. All traders, anywhere, have to follow the calendar of the market they are trading. E.g. If you are in UK, trading S&P in NYSE, then you have to know you cannot trade on July 4th as that is a holiday. If you are holding a position you want to sell that day, you cannot.
All trading systems are market aware, they plug into them holidays, settling currencies, settlement rules etc which local traders know and expect them there in their trading system.