I would fly into Heathrow, because you are then connected directly to Paddington Station via the Heathrow Express non stop fast train. Then get a hotel near Hyde Park, I favour the Thistle Hyde Park, but there are others right there. If you want really grand try the Ritz, near all of the places I mention. You will be near the Swan pub and the Victoria pub, both historic. The Victoria has three fireplaces. Take a cab to Holland and Holland, then pop across the road to the old pub right there. Then walk up to James Purdey and Sons, maybe 15 minute walk. Pop in at Barclay Bentley store on the way just up from the pub to smell the leather, and then the Porsche shop across the square.
Once you have had your Purdey fill retrace your steps, passing Churchill tobacconists, nice range of cigars. When you get back to the Porsche shop turn left, three blocks up is Gray’s antiques, many interesting things.
For WR take an early fast train to Birmingham, it is a short cab ride from the station. Contact Steven Haridence to organise your visit. Make it a relaxing day and end up back in London late afternoon.
For an interesting meal book the Rabbit in Chelsea, quite cramped but divine food.
Rigbys is quite a trek, expensive cab ride, but probably easier than tubes and busses. To fill the rest of that day before pub time maybe visit Farlow’s or Beretta at the top end of Germyn Street, opposite Davidoff. Your wife will like Fortnum and Mason department store near there, great high tea. Also a wander through Burlington Arcade, full of luxury goods shops. Also for the wife Patrick Mavros in Chelsea is an absolute must.