Help dialing UK cell phone number?

I am in America and am trying to send a gift to someone in the UK. The UK company with whom I am sending the gift needs a local phone number. I never make outgoing calls to this mobile number, but incoming calls come from 441753****** (* used for anonymity). I read somewhere that the local number comes from dropping the 44, but I also read that all UK cell phone numbers start with 7. This one would start with 1 if 44 is dropped. I am confused and looking for clarification. Thanks.