I will be travelling to the UK very soon and I don't want to change my phone numbers. There are certain very important calls I am expecting.
Would my US sim card (Virgin, AT&T) work when I'm the UK, so that I can still be reached on my US numbers?
If you want to receive phone calls, just make sure you have roaming enabled. You can call your operators to confirm, I believe sometimes you have to explicitly enable it. In countries I've lived in, enabling the service itself was always free.
Virgin for example provides instruction for setting up roaming in your account, similar thing should available for all networks.
Get set up – If you’re planning a trip abroad and want to use your phone while you’re away, you just need to let us know by switching on Roaming in Your Account. To do this, log into Your Account and select the Manage Services tab. From there scroll down to the Roaming section. Click the slider to the On position, and you’re all set.
Just be aware that prices even for receiving calls can be really high (e.g. for receiving my Polish phone in Brazil I'm paying 2 USD / minute).