It's definitely not too early. You may actually realise you should've started sooner.
Getting a temporary place (eg 3mo while the owner is travelling) to find something permanent is a plan that many choose to follow. Even so, you might end up getting a 2nd temporary place and so on... And finding a temporary place might turn out to be just slightly less difficult than finding a permanent one.
Boligportal.dk is probably the best place to start. The 2nd one would be dba.dk The problem with DBA is that it's free and everyone is constantly looking for places on it. Owners get spammed with literally hundreds of messages and phone calls within the first few minutes from posting; a lot of these messages come from automated scripts, even I had a browser extension refreshing every 5s and if that got me a reply, it usually was something like "sorry, you're number 15, I'll probably rent it before you get a chance".
Boligportal.dk is a bit less crowded because it costs a monthly fee, which might deter some crowd from joining, like students or people looking for the cheapest alternatives. So, Boligportal properties might be a bit more expensive, but higher chance of at least booking a meeting.
Speaking of which, the #1 reason why you should start asap is: students and the seasonality of renting. There's usually a high influx o students late summer and as you get closer to September everything gets more hectic and people more desperate.
As for prices, not sure what you are used to in London, but there's an obvious inverse proportionality between price and demand. Ideally you'd want a 1bedroom/condo for 7000 DKK in a central area, but that's so ridiculously hard to find and get. Two rooms might get you between 8-12000 DKK and that's a more accessibly market, but still crowded. If you're willing to go to 15000 DKK/mo you're probably going to find it easier. The next parameter to add to the equation is postcode, and anything =<2200 is generally within reasonable daily biking commute distance (not sure where your workplace is, though).
You'll soon add more params, such as proper shower (you'll understand), proper windows if old building (and they usually are), nearby shops, ideally a metro station, etc. But you'll probably start ignoring them if time passes and you're still in rent limbo.
Long story short: I strongly advise you take this very seriously and get busy asap. Go see every place you can, be highly presentable at the interview, bring a copy of your employment contract (with salary on it), manifest strong interest, really hope you don't smoke, fingers crossed.
I moved 4 times in 7 years and it was always nerve-racking. Last one took me 6mo; true, I was picky, but this is such a renter's market...
I wish you the best of luck. But as you already have a job, you might soon realise that finding a place to live is the next toughest challenge in Copenhagen