London, 2 days

will take express tube from Heathrow to Paddington station.

There is no express tube. There is The Heathrow Express, an expensive way of getting only to Paddington.

I suggest stay near Kings Cross/ St Pancras for an easy train ride to Gatwick. Get the Elizabeth Line or tube - Piccadilly Line - to get there,

Or take the Elizabeth line to Farriingdon. Either stay near there, or changes to a Thames link train and get off at Blackfriars or London Bridge. The Thameslink Train will take you straight to Gatwick.

We love the area around Blackfriars, and recommend Locke at Broken Wharf specifically. This is very close to St. Paul’s and walkable to the Tower of London, Shakespeare’s Globe (crossing the Millennium pedestrian bridge) and other attractions.

There is a list of garden ideas here. It includes some very well known ones like Kew Gardens (always a lovely place to visit) as well as less well known ones eg Chelsea Physic Garden.

I'd personally call the Elizabeth line from Heathrow to Farringdon an "express tube." It only stops once along the way.

The Elizabeth Line is technically a train, not a tube and it stops quite a few more times than once on the way to Paddington, so it’s not an express. Still quicker than taking a tube but not an express.

Yet another option is to take the coach (bus) from any of the Heathrow coach stations into London's Victoria Station area. You will arrive into Victoria Coach Station, just across the street from the train station. You might wish to stay in that area. Convenient area for getting around London--many tube lines run through there.

Then on your day of departure, take the Gatwick Express from Victoria Station to Gatwick airport.
It is an express rail passenger service. Buy your tickets in advance to get a good price.
Or you can take a regular (less expensive, slower) train from Victoria to Gatwick.

Even though you'd check out of your hotel after breakfast, you could leave your bags with the hotel while sightseeing and pick them up later in the afternoon/evening when you make your trip to Gatwick. (Another good reason to select a hotel very near Victoria Station.)

Hotels in the area of Victoria Station include a very nice Best Western, a Doubletree by Hilton, and The Victoria Clermont hotel. Just around the corner is the lovely boutique hotel The Lime Tree, very popular with our forum members.
https://www.limetreehotel.co.uk/

From London Victoria station you can also take southern trains to Gatwick. Slight slower than the Gatwick Express and somewhat cheaper.

Maybe some inspiration here?

https://www.visitlondon.com/things-to-do/openspace/garden/london-gardens

Hampton Court is beautiful on a sunny day, and a fascinating day out both outdoors and in. Also easy to get to with a train station opposite. My advice would be to get there as it opens and plan to spend a full day.

63 Bayswater BnB.

10-15 minute stroll from Paddington Station. Across the road from Lancaster Gate Italian Gardens in Hyde Patk.

Excellent pub The Mitre Paddington and a block & 1/2 away the small but wonderful Tudkin Flavors of Malaysia restaurant. These days probably need reservations.

As far as gardens check out Queen Mary’s Rose Garden, Kyto Gardens, and Kew Gardens.