Never had a Primark or Tesco shirt, but I can tell you that Tyrwhitt collars tend to be pretty bang on – they have a bit of room but aren’t massive.
As for the rest of the fit, once you’re up to 17″ and 18″ collars, the only sane option is to go for a slim fit shirt, unless you have some kind of incredible beer belly. If you’re reasonably fit and have any abdominal muscle, you don’t want the standard one.
My advice is to take a shirt you already have and fits well, then measure it up to find out what you want – that gave me a sleeve length about 2″ longer than what the bloke in Lewin’s reckoned, and meant my cuffs weren’t always around my elbows.
Tyrwhitt is streets ahead of Lewin IMO. The Lewin shirts I had were absolutely unironable, worse than some cheap’n’nasty Next shirts I had. Took them back under Lewin’s no quibble guarantee and bought Tyrwhitt’s non-iron ones. You still have to iron them if you want to look right, but it’s a pretty trivial job.
Neither make is really what I’d call super high quality – you’re certainly not getting an £80 shirt for £20 like they make out. The more expensive Osbourne shirts in Debenhams are made of good quality fabric, particularly the Pima ones, and are often knocked down to £20 or so.
A third option is M&S made to measure. IIRC they’re about £30, the collar came up a bit small, but the fit is very good. The standard fabric is relatively run of the mill.