So the check in agents for each of the three sectors for your return journey will really only be looking as far as your immediate destination to assess whether you have the right to enter that country. In case you are not aware although your first two return sectors are on BA they have a policy not not permitting check through onto other flights, even their own. So the check in agent at JFK will want to know you have the right to enter the UK at LHR, and they will not usually consider your onward flight to CDG even if your retimed to be the same day as your arrive. Additionally, if have check in luggage then you need to assume you have to go airside at LHR.

If you therefore opted to shift the date of your LHR CDG sector you may still need to assert you can enter the UK as far as BA are concerned. Also be aware that flight availability out of LHR is very limited due to a number of operational issues with airline and airport resources, so changing the date of LHR CDG might be difficult.
Although it's a separate issue, you also need to assert your right to enter Schengen for your flight to CDG.

My previous answer using https://www.gov.uk/check-uk-visa/ assumed you could direct transit at LHR which now appears to be wrong.

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