Welcome to the neighbourhood! Bayswater is lovely, and you're in a fab location for Hyde Park, the West End (including Oxford St and along to the theatres,) and you have a choice of tube stations to take you anywhere easily in London. (I am of course ever so slightly biased as I live here!)
All three of your closest tube stations will have at least one staircase to navigate. Queensway and Lancaster Gate each are served by a lift, but then have about 26 stairs down to the platform level. Bayswater also has a 26 step staircase from the ticket hall to the platform.
If I were travelling with my 81 year old mum, I'd take the Heathrow Connect (follow the signs for the Heathrow Express) for £10 each. It goes right into Paddington and is completely step free. At Paddington, I'd go to the National Rail ticket office (not the underground) and buy a weekly travelcard each as so to take advantage of the two for one free admission offers (you can also pick up a 2 for 1 booklet at the office). I'd then take a quick taxi to the flat or walk it if she's up for it.
Instead of going east from the Tower, you might want to head back towards Westminster. The Blackfriars is a lovely beautiful gorgeous pub located funnily enough at Blackfriars. For something a bit different, the Tattershall Castle is a permanently moored pub boat on the Embankment. If the weather is nice, you can sit on the deck and admire the view of the London Eye and Big Ben. On the south side of the river, the Founders Arms also has an outstanding view of St Pauls and it's near the Millenium Bridge and the Tate Modern.
As you're in the neighbourhood, go to Portobello Market on Saturday morning. Go about 9ish as it gets uncomfortably busy later on. The tube to Ladbroke Grove would put you in the middle of it, the tube to Notting Hill Gate puts you a short walk from the top of the hill to walk down, or you could take either the 7, or 23 buses from the corner of Queensway and Westbourne Grove and they'll take you to about the middle. For a very pretty gin palace of a pub, go into the Elgin located on the corner of Westbourne Park Road and Ladbroke Grove.
If you need groceries for your flat, there is a Marks & Spencer located inside the Whitely's shopping centre, and a Waitrose on Porchester Terrace. You also have a couple of small Sainsbury's located on Queensway as convenience stores.
Grainger & Co on Westbourne Grove do a lovely brunchy breakfast, but it's very popular and there will be a queue outside unless you're quick. Other good restos nearby include Ottolenghi and Bumpkin.