On my first day in the UK, I bought a pay-as-you-go Tesco Mobile SIM from the local Tesco supermarket for the handset I was carrying. It cost £2.49, including £1 of airtime. (Since then, the price has apparently dropped to £2.23, although this may be an internet-only price. See http://www.tescomobile.com/sim_only.aspx?pc=22&us=1&h=1)
Unfortunately the handset was broken, so I had to buy a handset. I didn't get around to it for about a week because of a hugely busy schedule.
However, I still managed to get a phone during a post-Christmas sale at Carphone Warehouse (http://www.carphonewarehouse.com). I got a Motorola C139 for £15, £10 for the handset and £5 worth of airtime on a pay-as-you-go Orange SIM. The C139 was simply the cheapest phone I could get. The cheaper Motorola C115 was £12.49, including £5 on the Orange SIM, but it was, and at the time of writing still is, sold out. (Both phones have now reverted to their pre-sale prices: £24.99 and £19.99, both including airtime. Unfortunately, they are also now both out of stock with the Orange SIM, although you could pay more for the C139 and get a SIM from another provider instead.)
Oh, another key selling point was that both of the phones from Carphone Warehouse were unlocked, which was important as I was going to be swapping in a German SIM on arrival in Germany.
I subsequently found out that the C139 has longer-than-average battery life, which is nice.
I decided to use the Tesco Mobile SIM as the tariff was cheaper for what I wanted to do, particularly because I could get half price calls and SMSs to three numbers I registered. I topped it up once, for £10, at a Tesco in Hammersmith. I used the Orange mobile SIM to call the States because I noticed there was a free credit of £3 one could get to call outside the UK, or to use the phone outside the UK, if you went into an Orange shop and asked for it. I went into the shop at Paddington and got my £3 credit, and burned a chunk of it before it expired.
Total expense: £27.49, including £16 of airtime, not including the bonus £3. Net expense for a handset and two prepaid SIMs: £11.49.
In Germany, I borrowed a spare SIM from my host. It's a Vodaphone CallYa SIM (€19.99, including €10 airtime). There are apparently a lot of different plans around. http://www.teltarif.de is a good comparison site. So far I haven't spent any money on the phone, since it still had about €4 of airtime left.