Perhaps we're looking at this the wrong way. Rather then "how effective would the French and Russians be?", perhaps we should be asking "how much damage could the Afghanis and/or Punjabis do, backed up by whatever limited assistance the French and Russians could provide?"
That view would certainly lead to a more interesting discussion. It also changes the topic from a 'Russia and French invasion of India in 1801', as the title suggests, to 'Afghani raid of India with a bit of a prompt from Russia and France'.
Punjab could cause a bit of trouble for a while, then the British would realise India matters more than distracting Napoleon, and the British would crush it. The Afghanis would be able to do hardly anything alone. A semi-nomadic raid that the Indian garrisons can take care of. Maybe a large part of those garrisons, but it would still be a raid rather than a proper invasion.
Would they? Maybe, depends on what commitments they have. Keeping in mind they tended to be a lot less complacent about invasion than pretty much anyone living today, I don't think they'd be so quick to give up on Europe, especially for a conflict that would probably be over before they could rush troops from one theater to another.
OTL Britain was at peace for a little while in 1802. Before then they had just broken Napoleons foray into Egypt. Napoleon wasn't doing a lot between then and 1805, so they weren't giving up too much anyway. Also, they had lost big-time in America 20 years previous, and what be horrified at the thought of losing the next most important colony as well. The Royal Navy was leaps and bounds better than the French Navy, so it can spare the stuff needed for the India operation. Everything else will be well above what is needed to protect Britain.
That definitely overestimates the strength of the East India Company. This is a corporation that, as you say, had half a subcontinent to plunder, and yet it could barely break even financially from year to year, had to be bailed out of bankruptcy by the British government in the past and would again in the future, and when they felt the need to go on the offensive against potential invaders forty years later, couldn't support more than 15000 troops for that purpose. I still don't think the French or Russians could get anything significant to India, but if they really could turn Ranjit Singh against the British, then that could get ugly. And it was less than fifty years beforehand that the British were in danger of getting run out of Madras, so they weren't that secure on the subcontinent yet.
I'll admit that I'm no expert on the BEI company or anything in detail about that time, so it is possible I am generalising a bit. However this was a time close to half way between the British first interfering in India (early 1700s?) and Victoria's title of Empress of India (1867). Something like the takeover of a continent doesn't happen in an instant, so the British hold on their half of India must have been firm enough that it could defend itself against the backwards remnants of the Mughals and other formerly important kingdoms. Technology must count for something, and the British were the masters of it at the time.
- BNC