Car starts first time..every time... in warm weather but will not start as it gets colder...45 degrees or so. What is the issue here?

 

Are the glow plugs functioning? Mine starts instantly as long as I wait for the GP light to turn off before I crank it. Beyond that there are people here with much more knowledge than me..

 

The Diesel Giant Website has a good pictorial on checking the Glow Plugs.
If you need an inexpensive Electrical Meter Harbor Freight stores sell them.

Other issues
Slow Engine cranking speed due to poor Cable or Ground Strap connections, Battery getting old (do an internet search and find a chart that shows what happens to your Battery vs Voltage when the Battery gets cold), Starter showing its age, thick Crankcase Oil (in the Winter synthetic Oil allows the Engine to crank more easily than regular Oil does)

Engines that have their valves adjusted have been know to start easier after the adjustment.

 

Single digits here in Detroit. Just bought my 1982 300D from a guy down in Louisville, KY. Has started like a champ every day for a month until this morning.... The glow plug light comes on and I can hear the relay kick off, but the car just will not start.... I have tried cycling the glow plug up to 5 times, and flooring the car and cranking the living daylights out of it but no luck. No idea what the issue(s) might be. The battery does not seem especially strong, no idea how long the glow plugs have been in the car, no idea on compression, even WD40 sprayed under the air filter did not do the trick. I am charging the battery tonight, but it is going to be even colder overnight. I might have to wait until it warms up a bit or get it towed to a shop. I had forgotton just how challenging it can be to own an old car as a "daily driver". Oh yeah, the car has some kind of heater in the upper radiator hose.... It looks like it should be in the lower hose.... Anyhow, I plugged it in and it seemed to do nothing.... It just drives me crazy. It is such a pain to work on a car in brutally cold weather.... Oh well, patience is the key. I will have to drive the Mini Cooper S in the salt and snow until I can figure this out. It might be a while as it is going to be below freezing all week.

 

DetroitFlyer,
I live in Buffalo, NY and have an 81 300TD. I had the EXACT same scenario last winter that you are describing, even down to driving my girlfriend's Mini Cooper in the mean time. It was working fine, and then one day no start, no matter what I did. I replaced the battery, I had the block heater plugged in, I did everything. It turned out that 4 out of 5 of my glow plugs were dead.
You can test them by checking their resistance. To do this, pull the 8 pin plug off of the preglow relay. pins 1-5 send current to your glow plugs. Connect one lead of a multimeter/ohm meter to ground (a clean spot on the engine block works for this) and with the other lead, touch each pin one at a time, noting the resistance reading. If the multimeter doesn't respond (indicating infinite resistance) the plug is dead or there is a break in the connecting wire. Each plug should read close to or less than 1 ohm of resistance. If you read over 1 ohm, replace the plug. Genuine Bosch plugs are cheap if you buy online, i think around $6-8 each if I remember correctly. They will be $28 each from a dealer. Whatever you do, replace all the plugs, even if you find one or two that still work.
Ohh, by the way, unless you are a very small human being, getting the plugs in and out without removing the injector lines is INCREDIBLY difficult. I did it at night, on the street, in single digit temps last year. It took about 3+ hours, and I had to have my girlfriend (who has very small hands) help me, because there was no way I could fit my big paws in there and turn a wrench on a few of them. Thank god for her little hands!
Good Luck!