Bath to Heathrow

General recommendation for any overseas flight is three hours, not two, although you will probably be okay with two.

I would travel the evening before - either a National Express coach direct to Heathrow or a train to Reading then coach, or a train to London Paddington direct from Bath then another train (the expensive Heathrow Express) back out to Heathrow. Then stay at a Heathrow hotel.

However I doubt that any of these public transport options are running at a time that would get you easily to Heathrow from Bath at stupid o'clock in the morning - really your only option at that time would be a car service. The drive would be perhaps 90 minutes at that time of day, so even arriving at the airport for 5am (which would be too late for my risk aversion) you'd be leaving Bath at 3.30am or earlier.

Bus? No, too early.
Train? No, too early.
Flight? No, there's no airport in Bath. There's one in Bristol but no direct flights to Heathrow.

Your options are, a very expensive taxi or drive yourself. At that time of the morning traffic will be light and should take around 2 hours

My option would be to spend the night before at one of the Heathrow airports. Having to be at the airport at 05.10 is not going to be pleasant so making it as easy and hassle free as possible is the goal for me

If it were me, I would have flown to Germany direct from Bristol Airport - which is just a few miles from Bath.https://www.skyscanner.net/transport/flights/brs/de/cheapest-flights-from-bristol-to-germany.html?adults=1&children=0&adultsv2=1&childrenv2=&infants=0&cabinclass=economy&rtn=1&preferdirects=true&outboundaltsenabled=false&inboundaltsenabled=false&oym=1708&iym=1708&ref=home

The Bristol airport plan sounds good to me too. There is a bus from Bath to Bristol airport.
Here is a list of all the places you can currently fly to from Bristol airport: https://en.wikipedia.org/wiki/Bristol_Airport#Airlines_and_destinations
BMI to: Düsseldorf, Frankfurt, Hamburg, Munich
EasyJet to: Berlin-Schönefeld
Ryanair to: Cologne/Bonn

As for check-in times, you need to get this info from your airline. Typical would be that you have to be at the gate 30 minutes before departure, and the check-in desk (only applicable if you haven't checked in on line or have luggage) doesn't open until 2 hours before departure.

The Bristol airport sounds like a great option to me. Otherwise, I think the coach from Bath to Heathrow and an overnight stay at one of the nearby hotels is the best bet. That's what I have done when returning to the U.S. after a stay in Bath and having an early morning flight out of Heathrow.

Perfect! The Bristol airport it is. They have several flights which make my deadline in Hamburg. Many thanks for the input!

bmi tend to be a bit pricier than many of the carriers - but you are saving that £120+ taxi fare. Book early.