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Unless you renew your visa, you would not be able to carry out actual work in the UK. This is because the business related activities that a person can carry out on a standard visit visa are not actually traditional “work”. If you refer to the heading “If you visit the UK on business” on the following link you will see this is the case:
There is no temporary form of visa that you can apply to extend your ability to work in the UK, the only visas available would be long term visas such as Tier 2 Sponsored Skilled Worker visas and this is not practicable in the time that you have left and the proposed length of time that you would work for them.
I am highly sceptical as to whether the entry clearance officials would let you back in if you left on the day that your visas expired. This is because they would see that you have until recently been living here permanently and they would suspect that your entry is not genuinely for visit/tourist purpose. It is probably more likely than not that when you attempted to re-enter from France they would refuse you entry.
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If you left the UK within 28 days then you would not receive a ban and when you came to re-apply in the future you would have to explain why you left later than when your visa expired but would probably be okay.
If you left within 90 days of your visa expiring then you would not receive a re-entry ban but the overstay would be taken in to account when you came to apply for a further visa to the UK in the future. It would make it difficult to get a visit visa in the future.
If you left after 90 days at your own expense then you would receive a 1 year re-entry ban, which means that any application submitted within 1 year would be automatically rejected. After 1 year ban has passed it would be very difficult for you to get a visit visa because they are permitted to take your previous overstay in to account even though the entry ban has expired.
In all cases, you must not work after your visa has expired. If you do then this would cause serious problems for you in the future.
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