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You can apply from the UK but clearly they will know you are an over stayer and you risk arrest.
But yes you can apply for a work visa from the UK.
I would suggest you use an agent or organisation to apply for the visa for you. Therefore they can deal with all the paperwork to give your chances the best success in the application.
Your employer is breaking the law as they should have a copy of your passport on file so would know the visa has expired.
Legally you can not keep your job while you apply.
Your chances are good if you can show that you can support yourself, you have stable employment and you have secured a life for yourself in the UK.
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