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The Museum of Lincolnshire Life does NOT have any records of the Lincoln Home Guard. There are some Lincolnshire Home Guard record in the Lincoln Archives, but none of note for the Lincoln Home Guard (I know, I have gone through the lot). You can apply for the records of individuals (if you are a descendant and you know their name) from the National Archives, but these records are very short, just telling you when they enrolled. The local newspapers (which can be accessed at Lincoln Central Library in person or the British Newspaper Archive online) have some articles that mention them during the War. The Northern Command list of officers has been published, but if you are looking for someone who is not an officer it will be of little use.
The book can be bought here:
https://www.amazon.co.uk/Home-Guard-List-1941-Northern/dp/1902366247

Good luck. There is a book on the subject going to be published in the coming year (hopefully) which might tell you more.