Does part 91 apply outside the USA?

Now let's take 91.175. The USA mandates the use of a published IAP.
The UK does not. 91.1 says p91 does not apply outside the USA. Does
this mean one can fly an unpublished IAP in the UK?

I don't see anything in 91.1 saying that it does not apply outside the US.
It says that it does apply within the US:

"(a) Except as provided in paragraphs (b) and (c) of this section and
§§91.701 and 91.703, this part prescribes rules governing the operation of
aircraft (...) within the United States ..."

and 91.703 then extends the compliance requirement outside the US "so far
as it is not inconsistent with applicable regulations of the foreign
country where the aircraft is operated or annex 2 of the Convention on
International Civil Aviation...".

So unless flying an IAP in the UK would somehow violate some UK rule, it
sounds to me like a US-registered plane in the UK must comply with 91.175
and use a standard IAP.

I take back what I just said. My copy of Jeppesen's "FARs Explained" cites
an FAA Chief Counsel opinion in which a US-registered plane operating in
Italy was not required to comply with 91.521 (shoulder harness) because
Italian rules did not require this. Similar decision concerning fuel
reserves, where 30 minutes (Italian rule) was acceptable in a situation that
would have required 45 minutes by US FARs. So the FAA interpretation of
"inconsistent" is apparently more lenient than my own interpretation. Hard
to believe!

I could not find this opinion online, the date is 2-16-1983.

A more recent (2006) Chief Council opinion (http://tinyurl.com/2h85xb)
concerning common carriage rules makes this less clear. The FAA refused to
say whether Part 91 was inconsistent with foreign law, and just said:

"The same regulations prohibiting the proposed operations under part 91 in
U.S. airspace, would also prohibit the proposed operations under part 91 in
Saudi Arabia if:

1) Saudi law is consistent with U.S. law (e.g. the Saudi government has
adopted the Federal Aviation Regulations); or

2) Saudi law is silent on the issues of common carriage and operations
conducted for compensation or hire.

The FAA offers no guidance on the Saudi requirements, but the U.S. regulations
prohibiting the proposed operations under part 91 are more fully explained
below."

So it sounds like if UK regs say nothing about a requirement to fly an IAP,
then 91.175 would apply, but if the UK regs state something different than
91.175, then it would not apply - and the FAA, if asked, would not give you a
legal opinion.