I've had a look at the IP addresses, and they all point not just to TalkTalk, but to the IP ranges that they inherited when they took over Opal Communications.
Opal was a business-only service, and is now called TalkTalk Business, but their IP ranges could be used by anyone now.
Although Opal/TalkTalk provide mobile Internet, all three IP addresses that elliew uses sepcifically say ""DSL"" in their lookup, which means that they are Internet connections provided through a Digital Subscriber Line - in other words, a telephone landline.
Whilst this doesn't specifically prove that she is lying, it suggests to me that she's accessing the Internet through a static point rather than via a mobile phone, and that static point has the same ISP at all times, which is unlikely if you're in different locations. One to keep an eye on.