Land LHR connect to LGW

I'm guessing these are two separate tickets.

Even though you didn't tell us where you are flying in from and which airlines you're flying. The simple answer is ....

NO. You would need a miracle to make it.

On a good day, it would take 1:15-1:30 to get from Heathrow to Gatwick. You will need two hours at least for check in and security at Gatwick. That leaves you 15 minutes to land at Heathrow, walk to immigration, go through immigration, walk through the terminal and find transport to Gatwick.

British Airways does sell some flight combinations that require a transfer from Heathrow to Gatwick. We did it 20 years ago on our way to Venice from Seattle. But we had 4 hours to make the transfer.

There is a National Express coach that does the route, with frequent departures from Heathrow.

https://www.nationalexpress.com/en/airports/gatwick/heathrow-to-gatwick

With 3.5 hours it will be very tight, and if your inbound plane is late arriving, or there are long lines at passport control, or terrible traffic, you could miss the flight. If all goes smoothly it should be OK.

I cannot find the link, but the official advice at one stage was to allow a minimum of 5 hours between arrival and departure times. You have significantly less - so it is highly unlikely you could make it.

Normally, when I have used Heathrow, I would consider myself lucky to get out of the airport in an hour from time of landing if I have baggage to collect. Eighty minutes has been more normal. A private hire or taxi might be able to do the journey in 45 minutes. so if the gods are with you, your flight is on time, you have a fast exit from the airport and the taxi is not delayed you might get to Gatwick with with slightly more than 90 minutes to spare to check in bags and get to departures.

I certainly wouldn't risk it on those margins.

NO - this schedule is way too tight.

The National Express bus will take a journey time of 1:30 mins between LHR and LGW. The train requires a change in central London and will not be much quicker. Your best bet is a private driver, but then you still have to deal with the M25 (notorious for traffic jams), I would say that will take about an hour if not more. You need a 5-6 hour window to be safe.

Looking at the TfL site, Heathrow to Gatwick, even using the Elizabeth line is still going to take you a minimum of 90 minutes - so not feasible as a part of 3:30 hour transfer. (I would still look for a longer transfer window, and use National Express)

It also depends which terminals the OP is using and how much extra time he would need for the journey (How much walking.)

With this much of a time constraint, I would forego the bus and either take a cab or a car service direct. It will save some time.

With just carry on and online check in, it might be possible. With checked bags...iffy.

If you already have the tickets, and they can't be changed, all I can say is "Good Luck"

All National Express Coaches stop at both Gatwick Terminals, so you need to get off at the right terminal as that might save you valuable minutes.

The Oxford Bus Company also run an hourly coach to both Gatwick Terminals. May be useful if you've just missed a NatEx., but an OBC one is due.
https://passenger-line-assets.s3.eu-west-1.amazonaws.com/oxfordbus/OXBC/LGW-timetable-20230119-d60402ee.pdf

I was just looking at an old ABC World Airways Guide from 1991. Back then the MCT from Heathrow to Gatwick was 2 1/2 hours rather than the 3 hours it is now.
Different times.
It was also interesting to remember that there were then Airbus express services from Heathrow to Central London, long gone, except at night when there is still a direct Nightbus.

And if I have checked bags on my first flight, should I assume, I will have checked bags on my second flight?

Not sure I understand the question. Why would you not have your checked bag(s) on the 2nd flight?
YOU are responsible for getting them LHR to LGW.

Remember, somebody has the last bag off the plane. It could be you. You could easily not be leaving baggage claim in less than 60 minutes. Then off to wherever your transportation is to get to LGW, travel time and getting to check-in/bag drop is about 2 hrs. This puts you about 3 hrs past scheduled landing time. Then through security, any border protocols to wherever the next flight goes and get to the gate all in the remaining 30 minutes.. ( insert Clint Eastwood's Dirty Harry movie clip asking 'do you feel lucky?')

I am with Frank that this looks to be a DIY 2 ticket journey, and join in his good luck wishes if you cannot change the tickets.

IATA minimum was still technically 2.5 hrs when I last looked at it but no airline has been daft enough to sell through connections in under 3 hrs for years. And that is pushing it in my view.

And if I have checked bags on my first flight, should I assume, I will have checked bags on my second flight?

Not sure what this is. If the question is when checked bags are transferred without collection then no. If it is about fees the you need to check ticket terms.

I did this transfer last December. However, I spent the night at Heathrow first so I could go to Christmas at Kew, so I have not done it in a rush.

I think you will need to collect your bags at Heathrow and transfer them to Gatwick yourself. We used the National Express bus that transfers between airports and leaves frequently. It was not obvious (at least to us) where to catch this bus in the terminal 5 bus area but we figured it out and had a pleasant journey. Traffic is unpredictable, like any metropolitan area and can impact the time it takes to make this journey. I did like the bus for ease of use and minimal luggage handling.

Is this all one ticket on BA? Is there a later connection you can make?

You definitely need to collect your bags and transfer them yourself, re-checking them at Gatwick.

But you have to go through immigration/passport control to enter the UK before you get to baggage claim, so your bags may well be waiting there when you get there.