I think it would probably depend on what your personal spending habits are, where you go t shop, how many magazines you read, how big of a sweet tooth you have, how often you plan on having lunch out, what exactly you plan on your pocket money covering, etc. Tim and I were just figuring out our expenses, and he realized he spends £40/mo on magazines (loads of train/model train/engineering mags!), whereas I only spend £10 (Doctor Who and the occasional women's interest magazine).
A cup of coffee and a pastry could run £8 or more, or be as little as £3 or less, it just depends on where you plan on stopping. Even a bottle of water can vary - you can get a bottle for less than £1, but you can also get bigger bottle sand fancy brands for as much as £5. Of course, you can save yourself even more if you pack a water bottle and pack a flask of coffee. The flasks we bought were less than £5 each at Tesco and they keep tea hot for well over 12 hours. The same goes for sweets, crisps, and other snacks. You can pick up at 5-pack of chocolate bars sometimes for as little as £1 (when they're on special offer), or you can spend 60p here and there each time you want a chocolate bar.
There are 4.2 weeks in a month on average. If you break down your £150, you're giving yourself about £35/wk. Many of us probably could live on £35/wk pocket money, but the question is, can you? My suggestion would be to budget yourself a little more at the beginning and keep track of what you spend each week. You're still in the US, right? I realize that prices for things will fluctuate, and some things will be more expensive in the UK, but £35 is about $58. I'd take $58 out in cash and see if it can last for a week.
Also remember that London is generally more expensive than the rest of the UK.