So on the one hand. That's really awful and sad and I would be keeping the dog because no fucker who cares an ounce it would leave it with someone for far longer than arranged and not even say anything. What if you had your own holiday or work commitments meaning you had to just put the dog in a shelter? It's incredibly irresponsible.
On the other hand, the crime junkie on me says, she is bothering to open and read every message you have sent, hasn't blocked you or left you on unopened, yet has not replied to any message for between 4 and 6 weeks.
You don't know where she lives so you can't physically see she is home safe, you don't have a sufficient social media relationship to tell whether she is genuinely posting herself, and you say you heard her talk once, briefly, on the phone, in a crowded or otherwise loud area.
Not many people (I'm sure some do though) spend between £2 and £3k on anything, let alone a companion animal, and just ditch it. Not many people can book a 2 week holiday but then disappear for 6 weeks.
If she was back home why wouldn't she pick the dog up? If she didn't want it why wouldn't she sell it to some other unsuspecting person or at least ask you if you want him? It's not hard to guess someone would get attached to the dog after 6 weeks. You've mentioned a potential vet trip but presumably the dog is not so ill it was worth mentioning in the OP so, presumably, he would be sellable.
Just some major major red flags in this situation. I agree with other posters saying I would send a message saying something along the lines of the following:
"Hi [ name ],
I'm sure you can understand that I'm a little concerned that you haven't made contact with me since you dropped Frenchie off. You advised you'd need dog care for 2 weeks but it's now been over 6 weeks with no word and I'm worried about you.
I have fallen in love with Frenchie over the 6 weeks we've spent together and I'm more than happy to keep him if you are no long able to care for him. However as anyone would be, I am worried about the lack of communication so please, give me a quick phone call to let me know you are okay.
If I continue to not hear from you at all I do feel like I should call the police to do a welfare check as I can see you are reading these messages, I honestly am just concerned about you at this point."
This is friendly, should cover all bases, and should help ease a response out of her if it is some kind of mental health crisis or difficult personal situation. It also gives fair and non threatening warning that you will contact the police if she doesn't get in touch with you.
P.s. Please attach dog pic for us 