southampton port to Gatwick (LGW)

Sounds like you're arriving on a cruise. Check with them-when we arrived at the port in July ,they had arranged buses to LGW and Heathrow. Don't remember if there was a cost or not.

I don't know your tolerance for stress, but it must be greater than mine. I would never try such a close connection.

You don't say when docking is, nor when they will let you off the ship.

What I do know is that for a transatlantic flight you will want 3 hours prior to departure for arrival at Gatwick.

So if your flight is at 2, that's 11:00.

The last train combination from Southampton Central that would work if the stars align would be the 8:30 to London Waterloo. Change at Clapham Junction at 9:42, you have 11 minutes to get from platform 7 to platform 13 (doable easily in 11 minutes) for the train from London Victoria at 9:53 to Gatwick, stopping just once at East Croydon. That train gets to Gatwick at 10:19.

The previous train combination at 8:00 gets in at 9:55.

The question is, how lucky are you? Will they let you off the boat in time to get a taxi to Southampton Central that early? Will everything run to clockwork on both trains, in Monday rush hour?

If you get a car service will they run to time, no issues on the motorways? Again, how early can you get sprung from the boat?

The fastest route by car is towards London on the M3, then onto the world's biggest carpark, the M25 to the M23 and down to the airport. On a great day that's about 2 hours, easily significantly longer. So better hope you can be in the car by 8:30 for comfort, earlier if things go a bit skew-whiff.

Same time really, same risky strategy, different risks.

I expect that others will be more up-beat than I; I just hate worry and uncertainty, and the cost of a new ticket if I miss the plane.

Taking public transport will be about a 2 1/2 hour trip from the port. A car would take around 1 1/2 to 2 1/2 hours depending on traffic.

I think it's pushing it for you to expect to get to Gatwick by 11 am. Even with no delays (what if your ship is late?), there might be other delays on the road. If it were me, I'd opt for a later flight, or even spend the night there and head back the next morning.

But it really depends on your own level of comfort.

Many of the people responding have never disembarked in Southampton. They are giving you the best advice they know how, but are missing the experience portion.

I’ve disembarked there twice. Both times people carrying their own luggage were coming off the ship by 7:30 am. I took National Express buses to Victoria station once and directly to Heathrow the second time.

I find Gatwick fairly user friendly for international flights. I think as long as you were at check in by 11:30 you would be fine for a 2 pm flight.

There is always a risk factor that some unexpected event could delay your ship’s arrival or disembarkation times. You would need to think through how you feel about this.

There are many taxis available at the terminal and most of them would like nothing more than a long fare to Gatwick. That is one option. A cruise line provided transfer is another, but it won’t get you on your way to the airport as quickly.

If this connection stresses you out, consider going for the 2 o’clock flight the following day. Spend a night in London and eliminate most of the transfer pressure.

There is always a risk factor that some unexpected event could delay your ship’s arrival or disembarkation times.

That is exactly the problem, along with the fact that you would have to self dis-embark (that is handle your own luggage) to make the time- not wait for the ship's disembarkation where they handle the luggage. Not everyone can self- disembark, only a small number.

But many things can delay the ship's arrival - earlier delays at sea due to weather, mechanical problems, assisting at an SOS, medical evacs. And you can't afford any delays. But you could be on time all the way to the Southampton Pilot Station then be delayed there. My worst delay (as a seasoned cruiser) was 8 hours coming into London Tilbury- when it was too rough at sea to embark a pilot, we eventually had to divert almost to Dover to get calm enough seas to embark the pilot. Coming into Antwerp I've encountered the pilot embarking hours late by helicopter- due to high seas off Flushing. I've had pilotage delays quite a few times round the world.
Also (and people on infrequent cruises may have not encountered it)- as a regular user of the Solent I have had a number of times when Southampton Port have closed the Solent approach due to fog conditions, so all shipping movements (cruise, ferry and freight) are stopped and you gently circle off the Isle of Wight or even change arrival ports- it does happen.
I would highly advise flying out the following day.

Many of the people responding have never disembarked in Southampton.

Not me. I did it and embarked too, many times, at one point just about every year.

But, to be fair, this was from Southampton Ocean Terminal and into Waterloo on the Cunarder Boat Train, and other boat trains, and many were hauled by steam locos. And a few years ago.

Do any 'international' cruises dock there or only UK originating cruises? i.e., is passport control an issue that needs to be factored in?

arrive on Southampton port on a Monday morning

What time is the boat scheduled to dock? What time do passengers get to get off?
Without that info, I do not see how anyone can provide reliable estimates of your prospects of getting to LGW by 11AM.

Do any 'international' cruises dock there or only UK originating cruises? i.e., is passport control an issue that needs to be factored in?

All cruises from Southampton are 'international' in the sense that they visit international ports with the very odd exception, but the vast majority are UK to UK cruises via continental ports. One of the interesting issues next year will be ETA. However cruise companies the world over handle passport and visa issues on board, transmitting that information electronically to the relevant authorities.
That data is collected when you embark a cruise at check in.
At Amsterdam, Rotterdam and Barcelona I have intermittently encountered passport control, but the UK hardly ever.
I can count on the fingers of one hand (with several to spare) the times I have arrived back in the UK from a cruise and any passport booths have been open for any nationality or that the customs lanes have been staffed.

What time is the boat scheduled to dock? What time do passengers get to get off? Without that info, I do not see how anyone can provide reliable estimates of your prospects of getting to LGW by 11AM.

Pretty much all cruises are scheduled to dock at Southampton between 6am and 7am. Those experienced in cruising know how a ship is debarked. It is also a known the time it then takes to get to Gatwick. It is an inconvenient fact that arriving by 11am at Gatwick requires a lot of luck, with nothing going wrong either by train or bus or by car. Yes self disembarking from an on time ship arrival gets you there on time with no issues arising. As said the ship could be late for many reasons, the trains could be delayed, there be an RTA or other incident on the motorway- when the latter happens surrounding roads jam up very quickly assuming you are not trapped in a traffic queue on the motorway. It's a high stakes game of roulette.

The real scheduled time of arrival (as opposed to what the cruise line tells the passengers) is there for all arrivals (freight, ferry and cruise) to see on Southampton VTS- planned and actual. The proviso is that the given times are usually at the Nab pilot station, not alongside for cruise and freight.