score:4
Check in should always be done with the airline actually operating the first flight, even in the case of codeshares, so here it is Jet Airways. If you show up at the United desk, they may accept your check-in, or more probably redirect you to the Jet desk.
I don't know what you mean by "partner airlines", but I see that Jet is not a Star Alliance member, so there is a possibility that you and/or your baggage will only be checked in up to London. This should be evident after you have completed check-in at Mumbai and received your boarding pass and baggage tag:
If you do not receive a boarding pass for your London-Newark flight, you will have to check-in again at London (this time, at the United desk since they are operating the London-Newark flight)
Likewise, if your baggage tag only mentions the Mumbai-London flight, you will have to collect it at London and recheck it.
I suppose that if you are checked in, your baggage will be as well, but I know that the converse is not necessarily true (that is, it is possible that your baggage will be checked in all the way but not you, it happened to me just this month albeit with different airlines).