In Thor: Ragnarok, we see Thor and The Valkyrie discuss about his returning to Asgard. She says if Hela is truly there then Asgard is doomed. She also went on to say "Hela draws her power from Asgard as well as you." Is this true and if so what is the case now that Asgard is destroyed?
Thor draws his power from Asgard. As Odin says in the movies Asgard was never a place but its people. Thus Thor draws his strength from all Asgardians. When Thanos killed 50% of the Asgardians Thor lost roughly half his power. That's why he sought out Stormbreaker to augment his power. As Asgard prospers again on earth Thor's power will grow once again.
As it has been stated in the other answers, Odin specifies that Asgard is not a place, but its people.
The difference between Hela and Thor was this knowledge of where exactly their power came from. Hela was obsessed with placing herself on the throne of Asgard, to become the Queen of Asgard. She believed (as did Loki, originally) this was the seat of "ultimate power". Thor, on the other hand was told that Asgard is just a place, and what good is a place, if there are no people? This, along with the lesson he was taught about Mjolnir (it is only a tool to help direct his power, not the source of it), was the difference between Thor and Hela. Which is why Thor was intent on saving the people, and Hela directed her attention toward Surtur as he was summoned to destroy Asgard, while the rest escaped.
Following the release of the final movies, we can see a significant difference in Thor's power. After Thanos carries out his wish, and "snaps" half of all life out of existence, Thor is slow, lazy and fat. After they travel through time, and "un-snap" everyone that was removed, not only does Thor show his true colors once again, but also has a significant increase in power. This is because the original population of Asgard has been restored.
Thor: Ragnarok was perhaps the least comic-accurate Marvel movie to date, however, given that the rest of the MCU has stayed pretty true to the comics, the answer to your question should keep with what we know from the comics.
In the comics, Thor is Asgardian, which means he possesses the above-human strength, speed, endurance, invulnerability, etc. based solely on his physiology. "Thor’s unique physiology of half-Asgardian and Half Elder God grants him the typical powers of an Asgardian God but he is much stronger than the typical members of his race" (Comicvine).
Here's a link to a more in-depth discussion of his powers: https://comicvine.gamespot.com/thor/4005-2268/
Since his abilities are mostly due to this, he should retain all of his natural abilities (including weather manipulation) even after the destruction of Asgard. Mjolnir did assist with some of his natural abilities, and even gave him some new abilities like flight, so with the destruction of Mjolnir, we may see a flightless Thor in Infinity War. But the destruction of Asgard should not have taken away any of Thor's abilities, since they are based on his physiology and not on the existence of Asgard.