Focus on the things you know.
You know: "no responses to emails (some that required a response), work not completed as agreed, and not completing their timesheet".
You are only "suspicious that they did not work".
Therefore, focus on the things you know for sure, and ask questions about those things to ascertain what was really happening. "You know you are supposed to fill in timesheets, why didn't you? Oh, couldn't access the VPN? What did you try to do about that? Did you contact IT? Why not?" (Adjusting the follow up questions accordingly).
Then: "We estimated X for task Y, so it should be finished by now. Why is it not finished? Oh, you discovered problem Z? Why didn't you let me know it wouldn't be finished on time? How did you try to solve it? Who did you ask for help? No-one? Why not? Please show me what progress you have made."
Then: "You did not respond to this email A about subject B from person C, though it clearly needed your response. Why not? You also did not respond to my text messages. Why not?"
(Presumably the answers will not include "I had complete failure of internet connectivity and phone signal" as they've been able to contact you to tell you that they are off sick. So whatever answers they give, keep asking "why?" until the situation is clear. Why didn't you do it? Why did that problem stop you? Why weren't you able to solve it? Why didn't you tell anyone about it, or ask for help?).
The idea is not to validate your suspicion that they "did not work", and not to make any accusations of which you are not already sure - you're not a lawyer and you're not trying to prove your case to a jury. You are simply bringing to this person's attention that there are at least three things (sounds like more) they knew were supposed to have done, but they did not do; and that unless the answers to your questions give genuine explanations otherwise, this is not acceptable.
On the assumption that they did not do what they were supposed to and have no good explanation, that sounds like a disciplinary issue; how this is handled in your company may vary and may depend on the laws in your area, so talk to your boss and/or HR first, but I'd say this probably qualifies for a formal written warning as to their conduct. But perhaps discretion can be shown, avoiding a disciplinary. Perhaps retrospectively call the missing two weeks vacation rather than working time, deducting it from their vacation allowance if they have any left; or agreeing to take it as unpaid time off if they don't have allowance remaining. Again, this will depend on your company, your judgement, the laws in your area, and the employee's circumstances.
Additional note: you should brace yourself for the possibility that they may not come back at all, or that they may resign shortly after doing so. I'm not saying this will definitely happen, I can't know that, but I've seen things like that happen before: someone takes an extended break; either because they're already unhappy with their job or they realise they're unhappy while on the break, especially if in that break they see family or their home country for the first time in a long time, and they check out. It can happen.