I've seen "work in progress" building papered houses sit for considerable amounts of time.

However I don't know how the weather affected the structure, long term.

I wouldn't worry about middle term if the paper was applied properly, overlapped and tacked on so as to remain weather tight. I wouldn't do it with a Tyvek type wrap but (if I had to) I'd spend my money where it would give me the most benefit.

Only you can decide that. Its not risk free to let it sit as is.

Finish the siding first, then scrape up the money to finish the interior.

Damage from unprotected exterior walls will quickly surpass the costs to properly protect them in New England winters.
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PS: Housewraps can be left exposed for a maximum of 90 days before UV damage renders them useless and they must be replaced.

I would side it too.

I've seen the other manufacturers of fiber cement lap siding offer good prices. I paid $65-70 a square for 6" exposure primed planks. I haven't paid attention to the shingle prices.

I am assuming you have no code/city/county inspections?

Without a protective surface the UV form the sun will make short work of the tar paper.

It gets brittle and is easily damaged after a short period.

Siding does not really stop rain from penetrating, the drainage plane behind the siding is what keeps out the water.

The siding is to protect the drainage plane and make things look nicer.

Typar claims their HouseWrap can be left exposed for 180 days compared to Tyvek which claims their HomeWrap can be exposed for 120 days.

But Tyvek claims that their own tests prove that Typar loses some degree of water resistance in 30 days and Tyvek does not but I wouldn't rely much on self-serving testing. It's possible the Tyvek testing used Typar made before 2003.

The commercial versions of these wraps are more UV resistant and perform better in most other ways as well so that is what I would recommend if the OP chooses to delay exterior cladding installation until spring. Typar claims that MetroWrap can be left exposed for 12 months and Tyvek claims that CommercialWrap can be exposed for 9 months.

Tyvek also makes a fluid-applied weather barrier and claims it can be left exposed for 9 months. In my opinion it (and other systems from STO, Grace, etc,) are far superior to any other form of weather/air barrier.

Also, some claim that Typar is more resistant to surfactants from wood shingles but Tyvek claims all air/weather barriers are affected. White cedar might not be as much of an issue as red cedar. Will the shingles be dipped in stain?

An alternative is to use asphalt-saturated felt and replace it before installing the cladding.