Yes Apple Pay will (technically) work practically anywhere in the world where you see the Visa payWave or MasterCard PayPass symbols, including on the London Underground.
However, there are a couple of things that you may want to consider before using Apple Pay abroad.
Firstly, as you have mentioned, informing your credit card company of your travel plans. This will ensure that your card does not get flagged or suspended for potential fraudulent activities when used overseas.
Secondly, as pointed out by Terence Eden in the comments above, double check with your credit card provider about fees and exchange rate premiums for foreign transactions. These vary widely by card issuer and in some cases can be exorbitant. As an example, one of my UK credit cards charges 5% exchange rate premium, plus a 2.9% foreign transaction fee. Your card company may market "fee free", but that may be masking a higher exchange premium.
I live in Hong Kong and have used Apple Pay with US and UK cards in over a dozen countries with no issues, including using a US card on London trains and buses before Apple Pay was officially released in the UK.
Although the limit per Apple Pay transaction in UK is currently £20 - you will be subject to the Sterling equivalent of whatever limit you have back home. With some merchants, you may also experience a slightly longer period to authorise your transaction.
Although Apple Pay was limited in the past at £20 then £30, there is now no limit on an Apple Pay transaction. There is a £30 limit on contactless card payments though.