This is both a legal and political question.
All three countries have extradition treaties with each other. Citizens of a country are usually more difficult to extradite from that country, but Assange is not a citizen of any of those countries. All extradition requests would proceed in accordance with these treaties. But these treaties can be complex: the US one with Sweden is a 20 page pdf, and the US one with the UK is a 147 page pdf—though this covers a lot more than just the United States, which is a "category 2 territory" in the terms of the treaty.
And there is always a political component, especially when there are multiple countries looking to extradite someone. If Assange was first extradited to the United States, as you suggest, then in accordance with what's known as the doctrine of specialty the US could not then extradite him to Sweden unless they were given permission to do so by the UK; nor would Sweden extradite him to the US without permission, if the order was reversed. Negotiating such permission would be a political matter between the involved countries.
Additionally, Ecuador, in revoking Assange's asylum and handing him over to British authorities, required that the UK promise not to extradite him to any country where he may face torture or the death penalty. This could be viewed as a veiled "do not extradite him to the US" demand, as the US both has the death penalty and an apparent willingness to torture people (especially those foreigners associated with the events Assange is also associated with). The US could, however, promise the UK that he would not face these dangers, and as long as the UK accepts these assurances then they could extradite him to the US without formally violating their promise to Ecuador. Though whether or not Ecuador or other nations would see it that way is another matter. These are in fact common conditions, and many countries which lack a death penalty will refuse to extradite someone to a country where they may face it, even if they accept the accused is likely guilty.
So while matters will proceed according to the treaties, where he gets extradited, if anywhere, and in what order and under what conditions, will involve a series of political negotiations and calculations.