US citizen in UK on a tier 4 visa. When it expires I want to leave & then re-enter UK as a Tourist, do I need to go back to US first?

My question is whether I have to go back to the US to re-enter, or whether instead I can travel to somewhere closer (e.g. France) before re-entering as a tourist?

Nothing in the immigration rules preclude what you’re planning to do. Several people have returned to the UK after a brief trip to adjoining countries. That said a visitor in addition to demonstrating ties to their place of residence must show

Genuine intention to visit

V 4.2 The applicant must satisfy the decision maker that they are a genuine visitor. This means that the applicant:

(a) will leave the UK at the end of their visit; and

(b) will not live in the UK for extended periods through frequent or successive visits, or make the UK their main home; and

(c) is genuinely seeking entry for a purpose that is permitted by the visitor routes (these are listed in Appendices 3, 4 and 5); and

(d) will not undertake any prohibited activities set out in V 4.5 – V 4.10; and

(e) must have sufficient funds to cover all reasonable costs in relation to their visit without working or accessing public funds. This includes the cost of the return or onward journey, any costs relating to dependants, and the cost of planned activities such as private medical treatment.

In your case your reason for reentry so soon has to be tailored to satisfy this because unless you’re coming for something very important and time dependent (e.g. wedding of close friends), the question would be why didn’t you undertake those activities while you were a student? Visiting family for example could reasonably have been undertaken during breaks from school.

Going back to the USA for a very brief period and then seeking to reenter will be of very marginal value. The best cure for your situation is going back home (USA) for an extended period and thus demonstrate ties to a location other than UK.