The problem you'll have in extracting heat from your exhaust gases to heat your engine is twofold:
Gases don't generally have a high specific heat capacity so though it's hot, it's not a lot of heat energy, and it'll take some time to heat up a large mass. The walls of your exhaust pipe are incredibly thin, and quite effectively insulated by the air around them so they will indeed heat quickly. If you took a fan and blew air over it to warm the air, you might be surprised how quickly your glowing cherry red exhaust cools down again. If you wrapped the exhaust pipe in a 2 inch jacket of water, it will take much longer to heat this contraption than it would the bare exhaust pipe
The condensation caused by actively cooling the exhaust gases would be considerable and it's quite corrosive to the metal of your exhaust. You might find that it accelerates the rate at which you have to replace the heat exchanger or exhaust, and the associated cost of doing so makes it an economically unviable option
Investigate whether your heater system is functioning properly; if your thermostat is stuck (partially) open (as they're designed to when they fail) then it will take a lot longer for your engine to reach operating temperature.
It's in the best interests of fuel economy (and hence manufacturer reputation, so there's a market drive for this) for an engine to warm up quickly, and they're thus designed to have a really small cooling circuit (the engine block, and the heater matrix/exchanger that heats the cabin via the vents) thanks to the main thermostat being totally closed when cold, causing the engine to heat quickly, before the main radiator starts being used to dump the excess heat.
Consider too, running the blower at a lower speed so you don't suffer human factors such as windchill meaning you perceive the air from the vents to be cooler than it is, because it is moving at high speed. Run the recirculate mode, so your heater is not drawing fresh cold air from outside, but keep the AC on to dehumidify the air, especially if you're running it in recirculate rather than fresh. Excess moisture on surfaces inside the car will sap the heat from the air, as the evaporation of the water needs to get its energy from somewhere - running recirc without AC will mean that over time the moisture in the fabric of the car interior and on its surfaces will increase. You'll be wiping the windshield every morning with a rag so you can see, etc..
Alternatively, it's possible to obtain (in my country at least) aftermarket electrical accessories such as heated seat covers and fan heaters that plug into your cigarette lighter socket. It's also possible to equip your car so you can plug it into your house energy supply overnight to preheat the cabin and engine with electrical resistance heating (cold countries find such systems a necessity even for getting the engine to turn over in a morning). Keeping it inside a garage may help too as it won't go so initially cold.
If you're after a cheap and easy experiment of a similar ilk to what youre proposing here, consider ducting the air from around the engine, particularly on the exhaust manifold side, into your cabin air intake instead. Older caburetted cars often had a cowl around the exhaust that was part of the air intake to the carburettor, controlled by a bimetallic strip, with the aim of reducing temperature variations of intake air (reducing condensation of fuel on the surfaces of the intake) and make it require less choke in early running; this system would be similar, but doesn't have so many condensation complications as stripping exhaust gas of its heat
Lastly, if you do go for the processing-exhasut-gases route and take it to commercial production, it could actually have considerable benefits for future exhaust design; if you can get the exhaust gas temperature down sufficiently that exhausts can be made of plastic, the condensation worry goes away.. but then cars would only need one exhaust to last their lifetime and the exhaust companies would buy and bury your patent </conspiracyTheory> :)
Edit: from the comments that indicate the goal is to warm the engine faster
You'll have a hard time getting exhaust gases to do this, for reasons of the aforementioned lack of serious amount of heat energy embodied in the air. Sure, if you could keep the gases in contact with the block longer then it'd warm slightly faster, but I don't think you'd see an appreciable difference on a short commute
There are crazy approaches you could take, like insulating your engine with spray foam, adding a very large thermos vacuum flask type thing to the coolant system (some luxury cars have this built in to provide cabin heat sooner on cold days), adding some kind of fuel burning device to heat the coolant circuit as you've noted. You could even buy a smaller engined car, or if your commute is short, an electric one (like a Renault Twizy) then you don't have to worry..
..but overall, I think you should stop worrying about it. An engine will have reached a reasonable operating temp long before the needle on the coolant temp begins to rise. Those coolant gauges typically have a very narrow range, like 90 to 110, with operating being 100. The parts doing the serious work will already be hotter than this long before, and it's the increment delay in heating your coolant that taking the time. It's not the cause that an engine had to reach a point where every part of the block is at 100 degrees before it's running efficiently and will be worthy of turning into an f1 car one day (BMW used old saloon engines to base f1 engines on, as the theory was they'd been through many heat cool cycles and the metal would be more free of internal stresses) so the short answer would essentially be "don't worry about it" - yes, some things might get knackered sooner but generally cars are so cheap and throwaway by the time they reach the "double the ten year lifespan they're designed for" point, that they just get scrapped. Running your engine up to full temp for every commute isn't going to stave off that death by much, "its old-age that kills cars, not mile-age"
Won't hurt to take a longer high speed blast a few times a month, especially if your car has a cat, diesel particulate filter or similar emissions control system that requires a good blast through, but don't worry about it having to be every journey; my commute shortened drastically from 50 miles to 1.5 five years ago, and the 04 diesel Volvo I've had for 12 years still behaves the same.. I don't make any special efforts to blast it regularly, but when it gives up irreparably I'll scrap it and hope to get an E60 M5. I firmly believe by having this dream my current car will run forever, just to keep it from coming to fruition :)