Where to purchase SIM card

Do a web search for "Carphone Warehouse" and find the one nearest your hotel. I found they had a better selection and much lower prices than the dealers selling SIM cards at Heathrow.

...and the people there will install it, activate it, verify it works, and tell you how to top it up if applicable. Much better than grabbing one at a kiosk.

Get your SIM card at Heathrow or order it in advance.

On my last trip in August to the UK, I went with a company called Gift Gaff.. www.giffgaff.com. I ordered their SIM card in advance and it took 10 days to arrive to me in the US. Two or three days before my trip, I exchanged SIM cards in my iPhone and activated the card online and chose a cheap (£8) monthly plan plus added £10 goody bag (for charges not covered by the plan.) Once I saw that it was working in the US, I popped the SIM out and stored safely in a contact lenses case and taped a paper clip onto the case.
Once I got to Heathrow, while waiting for my luggage, I took the US SIM card out and put in the UK SIM and restarted my iPhone and it all worked!!! Worked fine all over the EU, so it should work fine in Germany.

I told another person about Giff Gaff, and when they arrived at Heathrow, they went to the shop called SIM Local in Heathrow Terminal 5, next to the car rental desks and bought a GIFF GAFF SIM and had them install and activate it.

So that's two suggestions that might work for you.

We bought one from a Vodafone shop near our tube station. Very helpful and knowledgeable staff. There are stores all over, so just use google map or google search to find one that is convenient for you.

I used a GiffGaff card in the UK too.
Super easy; if you look at my past posts you'll find detailed instructions on how to order it, install it, and get it working.
It was great for maps on the go; I used the CityMapper app in London.
However, if you want to be sure to use the sim in Germany too, you'd be better getting one from CarPhone Warehouse, where they will advise you what's what.
Perhaps a VodaFone one would be better for you.

Thanks, everyone! I appreciate all the ideas. I'll take a look at giffgaff but think I will most likely find a Carphone Warehouse to purchase from.

We went to a Carphone Warehouse after landing at Gatwick in August and the guy told us to go to a grocery store next door, so we did and found a whack of SIM cards sitting at the till, They cost like 75p each but we had to install them ourselves. It wasn't terribly difficult but we would have preferred for someone to take us through that and I would have paid for the privilege.

In the end we got EE Sim cards, they were very cheap, used them in Paris for a few days too and it was pretty inexpensive for all the data we needed.