When someone wants to apply for a Standard Visitor Visa, there's a single appointment needed: the biometric enrolment at a VFS. Depending on the country, the applicant may have the opportunity to upgrade their application to 'priority'.
Applicants in the United States get an additional option, they can use a 'premium service' (cf 'priority'). These are sited at British consulates and have the equipment to enrol the applicant's biometrics and review their application (to help ensure its success) and pouch it to the British Consulate General in New York for a decision.
Having that as background, let's take a look at the form itself...
Critically, note the use of 'or' in the explanation. It's clear that a single appointment is required.
The same single-appointment process works for India, except there are no provisions for a 'premium service'. In fact, anywhere in the world, applying for a UK Standard Visitor Visa is a matter of a single appointment where the applicant's biometrics are enrolled.
Your questions...
The nearest application support center is two states away (Milwaukee,
WI) for me and the nearest visa center is also two states away
(Chicago, IL).
Pick Chicago if you want the premium service otherwise pick one of the DHS facilities.
Since I am travelling to India ... I was planning on doing my UK visa
application process then. Does this raise red flags?
They don't care; it will not raise 'red flags' and it will not be treated as exceptional either positively or negatively. Honestly, they don't care. Depending on which Indian VFS you apply through, you may be able to qualify for 'priority', but as explained there are no 'premium services' available in India.
Notes
- An applicant for the Standard Visitor Visa can apply from anywhere in
the world (given that a British consulate exists).
- A description of some the various premium services available in the
USA are here, those not listed are available on a case-by-case
basis.
In the special case for applications in the United States, the
applicant reports to a DHS facility where their biometrics are
enrolled. They get a receipt. The applicant then prints out their
application and encloses the biometrics receipt along with a
photograph(s) and passport. The whole bundle is then posted to the
British Consulate General in NYC.
There are also 'premium services' at Heathrow, etc for
'leave-to-enter'/visa-on-arrival applicants.
- A further thought on appointments and as a rule of thumb, if the consulate asks you to make a personal
appearance for an interview with an ECO, you should consider withdrawing the
application at once.
