No, and no.
And there have been more recent posts than 2013, I recall seeing one only last week. When you search a TripAdvisor forum the sort order defaults to "Relevance" and you need to change it to "Date".
Quiskest is likely to be DLR/tube and thi si also the cheapest, particularly if you use a contactless credit card to pay.
Rather too often a travel agent (occasionally the traveller and sometimes even the airlines) will route a passenger onto LCY and out of LHR in the belief that as its all London the transfer is quick and easy. It isn’t! BA did it to a relative of mine travelling with two children under 5!
It’s a pig of a connection there is no quick way to do it, there’s not even an expensive easy way.
Your choices are either DLR to central London and Tube to LHR or an expensive, slow car trip - perhaps someone will correct me if there’s a helicopter or speedboat alternative.
If you can change the flight into LCY to LHR or move the LHR flight on a day, book an hotel and spend some time sight seeing on London. Other than that you are stuck with a painful connection on public transport.
Sorry, but LCY is targeted at business travellers (and it’s well suited to that), it works fine if you intend to stay in London, particularly if you choose the eastern areas, but as a connection hub it simply doesn’t work.
Where are you flying from and to? Given that you usually need to be at LHR at least 2 hours before your flight that means 10.20, your chances of doing that are minimal. Allow at least 30 minutes to get from plane to DLR, change to District Line at Tower Hill then to Picadilly Line at Hammersmith. This is the cheapest way, TFL givea route using the Heathrow Express which takes 1 hour 6 minutes - that is the absolute minimum the journey will take.
It won't take 30 minutes to get out of LCY - that airport is really slick at getting passengers off planes quickly and bags off by the time you've cleared immigration.
Bag drop at T5 for the second flight will close at 1135. If you are delayed arriving into LCY, speak to an agent there and then - to ensure they rebook you into a later flight.
Use the DLR and tube - cheapest and fastest.