In itself, the shutdown wont really affect the outside world in the short term, border controls are still working so anyone travelling to the US will still get in.
The time it might start to affect us is in the medium term as taking the spending power from public sector workers might see a dip in the US economy which will affect their ability to buy from us. Such uncertainty also weakens the Dollar on the international markets which makes our exports to the US more expensive, the US is a significant market for the UK so there is a danger it could affect our recovery.
The big thing the pessamists and apocalypse peddlars are talking about is what might happen when the US hits its self imposed debt (and in effect borrowing) ceiling on the 17th October. This might result in the US defaulting on a bond it issued a year, two years or even 10 years ago (this is incredibly unlikely to happen). That would mean that puts all other US bonds into doubt, investors often use these bonds as security to raise finance - if your security and ability to get your money back is suddenly in doubt you become a bad risk and borrowing becomes super expensive. This has the effect of putting interest rates up and in reality, this is probably the main effect we would see over here - it might be quite a sharp rise in interest rates.
In reality, a bond default by the US is EXTREMELY UNLIKELY TO HAPPEN as they would cut spending on another part of their budget first before defaulting on foreign debt.
So, whilst at absolute worst case scenario it is scary, it is so unlikely you can probably stop hiding under the bed
. The governmental deadlock is unlikely to drag on that long and they will probably just vote a bigger debt ceiling.
It is an unseemly kerfuffle and one which shows politicians in their worst light but apocalyptic - no