We have seen problems with bad coolant temp sensors; cause this type of issue. On cold start up, you will get a rough idle and when revving, the engine can stumble with a rich condition (smoky exhaust and unburnt fuel smell). After the engine gets to normal operating temps; the engine will smooth out, regular idle and hesitation will disappear. Most of the time, a bad coolant temp sensor; will throw a trouble code but not always.
It is a good idea, to keep an eye on the coolant temp light in the speedo and see if it acts "normally". On cold startup, you should see a blue light and when it gets warm, the light should go out. There should not be any other type of modes shown.
Here are the modes of the coolant temp light:
Blue coolant light = engine cold
Flashing blue coolant light = fault in the coolant monitoring system
Flashing red coolant light = low on coolant
Steady red coolant light = engine overheating, pull over and stop engine IMMEDIATELY and do not run until the problem has been identified.
The might be other issues, that are fuel delivery or ignition related; could be a possibility, dirty throttle body (clean and do a throttle body alignment procedure recommended), low fuel pressure, failing coil, old plugs, etc. etc.
It is best to have a scan tool; that can show live data and that way, you can monitor, various things, such as coolant temps, maf readings, fuel trims, pending codes etc.
As for your MAF replacement; why did you replace it? Was there any related trouble codes; that would indicate it was faulty? It is best to monitor the MAF readings and that will indicate, if it is reading in spec. A replacement MAF, should be a oem BOSCH branded part; not a cheaper aftermarket MAF that will not work right, causing the car to run badly. If you have the old maf; you might reinstall it and see if the run any differently.
On these modern cars; a scan tool is crucial to see live data and note how things are running, that way you can see if anything looks off (e.g. fuel trims, maf readings, coolant temps, trouble/pending codes, etc.).
Fuel Trim Info - Ross-Tech Wiki
Start looking at your readings (fuel trims, maf, etc), checking for pending/trouble codes, check for a dirty throttle body, condition of your plugs, swap in your old maf and we can go from there.
A UK based service manual that covers your 1.6L engine; would be a good thing to purchase and that way, you can go over testing, troubleshooting procedures for possible failing parts. We do not have the 1.6L in the North American spec New Beetles; so, we do not have service manual info to refer you to.
The Haynes UK New Beetle service manual; seems to be a good and affordable option.
VW Beetle Petrol & Diesel (Apr 99 - 07) Haynes Repair Manual (Haynes Service and Repair Manuals) Paperback – 13 May 2013
Note: covers your engine: Hatchback, Saloon and Coupe, Petrol: 1.6 litre (1596cc),
ISBN-13: 978-0857336545
https://www.google.com/search?ei=gS...8459.2.2.0.0.0.0.121.237.0j2.2.0....0...1.1j2.64.psy-ab..0.0.0....0.nXXJoJS2CnQ