No, because this is not in any way going to cause consumer confusion, but instead is simply used for compatibility purposes.

Confusion is the measure used when trademarks aren't registered and is much harder to prove. When considering registered trademarks it's not necessary for confusion to be actually present at all - 100% of customers can know that your product isn't an iPad but if you advertise it as an iPad then you're infringing: it doesn't say "AppleWebKit compatible" in that string and the string is not required to make the browser function at all.

If, for example, people offer alternative services to people with an Apple UA string (BBC iPlayer did this) then you're impersonating such tech use via the trademark - it might be now for example that people need to turn off specific iPad enhancements as they break IE11 - that's commercial interference using a trademark. The trademark isn't needed in any technical way for IE to operate as it should, so it's not a compatibility issue.

Website content owners can choose to cut out certain browsers if they wish, impersonating another browser is fine .. but if you do that as a company utilising someone else's trademark to do so .. it just has feeling for me of not sitting quite right with what I know of TM law.

I would think the courts would recognize a difference between advertising to the end user for the purpose of purchase/consumption, and advertising to 3rd parties as part of a compatibility protocol.

The UA string is not visible to the consumer, and wouldn't affect the number of people who download and install the browser.

Its only purpose is to ensure compatibility with the server after the browser has already been installed.

Also I wouldn't say the browser is actually impersonating another browser. If it was trying to impersonate, the UA string would need to match the other browser exactly and it doesn't attempt to do that. The string is still uniquely identifiable as an IE11 UA.

Is it any more of a trademark infringement than 'IBM PC Compatible'?

Sega and/or Nintendo tried to require games to display their trademarks before their system would run them, courts ruled that third parties could use the trademarks to overcome such systems.

Will anyone be tricked? No. So not infringement.

Infringement, as far as I know is a civil matter. Who is going to sue when everyone is doing it?

I hope not. The last thing we need is a new round of lawsuits. Maybe everyone can agree on a set of rules for communicating this data.