Dose anyone know which airlines will let you fly a 4 month old puppy into the UK from Germany.

Lufthansa if it's a small puppy

You might find it easier for the pup to go over on the Chunnel and drive to pick up and come back, rather than flying.

I'd imagine most airlines would have no problems flying a puppy of 4 months.    I took one of mine, at 4 months from Gatwick to Geneva (as I recall I used BA but it may have been Swiss Air).   I've also imported two, different occasions, from the UK to Canada, using Air Canada - one was 5 months and the other 7 months.  Yes this is the 'wrong way' round, but I can't see you'd have any problems from Germany - PROVIDED of course, all the DEFRA (Pet Passport) is in order so he can come straight in.    I tended to use the airline of destination as from experience, they seemed to clear customs faster.   On no occasion did they fly other than in cargo.

I might be more inclined to go pick up the puppy direct from breeder in Germany, by road/tunnel however.

From what I've been told, it would be not only easier to cross the channel and collect the pup yourself, but also considerably cheaper for you once airline fees, import charges etc are allowed for.  I was briefly looking into importing a pup from the US and was told that even then, having them flown to the continent then crossing to collect was still hugely cheaper to do.

They can only fly into. UK as Cargo,  and then there are the quarantine station and clearance charges at Heathrow,  doubling the  cost.

As said cheaper to drive over and ferry or tinnel back.

Really only as cargo ? That isn't what happened to a friend & his assistance dog. His dog travelled with him in the cabin, not in the cargo hold. This is since the rules changed, when did they change back to having to travel in the cargo hold ? His hearing dog is only small & was allowed to travel sans crate as he is fully trained. There was also a puppy in a travelling crate on the same flight

As you say your friend travelled with an assistance dog,  which has to be from a recognised organisation like GDBA or Hearing dogs and one or two others.

This would not spply to a puppy bekng imported,  as per OP.

My friend's dog is a certified US hearing assistance dog & the puppy was a pet being brought home to UK by owner. There are multiple certifying organisations for assistance dogs not just the UK ones & of course the US hearing dog scheme predates the UK one.

Just wondered when they changed the rules again back to before the Pet Travel Scheme. Neither dog went through the quarantine centre, the passports were checked & being in order allowed into UK. Both dogs travelled in the cabin, the puppy in a travelling crate & the adult with it's owner. Must have been fairly recent that the rules changed back to pre Pet Travel Scheme as this was earlier this year.

To bring a puppy or dog into the UK, not an assistance dog they can't fly in the cabin on any airline.
You might find you can on a plane you charter yourself which is mega bucks.
Best bet is to either go over and collect direct from the breeder and drive back home or have it flown into Brussels or Amsterdam airport and collect from there. A courier company could do this on your behalf and would still work out cheaper than flying direct into the UK. Flying into the UK you not only have animal reception charges but also the airlines add a tax on too. Each airline will be able to advise what that is and you have to pay when you collect your dog not in advance.

As others have said, any airline will fly a dog into the UK but be aware that the UK only allows dogs in the hold (obviously except assistance dog). They can be charged as excess baggage but they must be in the hold. Animal reception fees are around £250 in Edinburgh, more than that at Heathrow and you will also have to pay vat as you would when importing any goods. A friend who imported into Heathrow had an unexpected bill of over £500 before they would release her dog.

It's much cheaper to fly from Germany to schipol and then get the ferry home, no import fees or taxes that way, or drive round the tunnel, it all depends where you live wheat the best crossing is.

" I took one of mine, at 4 months from Gatwick to Geneva"

I should have said he travelled in the hold, not with me in the cabin.   But by 4 months, puppies of my breed are not small

Could the breeder not meet you at the docks/tunnel in France, if you don't want to drive that far. You could take the tunnel over to France and meet them there? Personally, I would not want to fly a dog in unless I really had no other choice.

If you change your mind about flying and choose to drive instead please be aware of the laws in Germany & & other places regarding 'winter tyres' & 'snow chains', it is illegal in Germany to drive on snow or ice covered roads unless you are using both 'winter tyres' & 'snow chains'.

See the list of countries & laws below the writing on this michelin site:
http://www.michelin.co.uk/tyres/learn-share/buying-guide/ever-heard-about-winter-tyre-regulations

German Court Ruling 2010
https://www.german-way.com/travel-and-tourism/driving-in-europe/driving/snow-tires-winter-driving/