ETA: There seems to be general agreement on that. The question was asked, however, what is the legal position with using an existing company's IP for your own use? Any ideas?
As far as IP is concerned, commercial names and logos remain the property of the business concerned even if they're historic versions, and companies can and do start getting arsey if they hear that someone's using their name on something without permission. Some are maybe more relaxed than others, especially if it's not on some sort of profit-making venture (I'm guessing that the reconstruction and eventual sale of these leviathans doesn't keep you in Dom Perignon and Rolexes?); but because Boots UK is now owned by giant US conglomerate Walgreens, I'd have a care since giant US conglomerates aren't exactly renowned for a breezy, carefree approach to IP law.
It's probably worth contacting Boots' Head Office and asking to speak to someone in their communications and marketing team. Emphasise the local and family connection; stress that you're not going to be using their name in a pejorative way or to pretend you're in any way a Boots employee, just an enthusiast who wants their project to look historically accurate.
Any business with half a clue would be all over this like Billy Bunter in Asda bakery aisle when the 'final reduction' yellow stickers appear, as a golden opportunity for free positive publicity, the chance of borrowing it for promo purposes once complete, connecting with 'heritage' and 'core values' and all that marketing-guff, etc etc.
If you're really lucky and they've got community engagement cash swilling round unspent for this financial quarter, they might even chuck you a few pennies for the paint.
Unfortunately, not all businesses have half a clue... so success is not 100% assured. But the old phrase "better to beg forgiveness than to ask permission" may well work for some situations, but most assuredly does not cut a great deal of ice in a registered trademark lawsuit.
But hey, if you try approaching them and they don't go for it, then how about a very similar livery for the totally fictitious 'Poots The Chemist'...
(Source: Datsuncog Minor is a commercial solicitor for a certain high-profile Oxford Street retailer, with a particular focus on Intellectual Property law. He is not, it has to be said, a shiteist.)