In normal circumstances, with the two flights on the same ticket, this would be perfectly fine.
You're coming from the UK, which is not part of the Schengen Area, and heading to Peru, which is not part of Schengen either, so you will not go through immigration/passport control at all: you are staying in the "international zone" of the airport.
You may possibly not even have to go through security but I don't know if that would be the case in this specific instance. If you do, it's a separate security checkpoint from the general one for people originating in AMS, and those checkpoints are usually less busy in most airports.
So it's just a matter of getting yourself from one gate to another, while staff move your luggage from one aircraft to another.
Plenty of time to do that in 1h20, and most airports have minimum connection times (MCT) well below that.
However, these days both Manchester and Amsterdam airports suffer frequent issues, with all sorts of delays and cancellations.
So it's quite possible any of the following may happen:
- There are queues so huge in MAN that you cannot check-in/get to your gate in time and miss your flight
- The flight MAN-AMS is delayed more or less severely
- The flight MAN-AMS is cancelled altogether
- The flight MAN-AMS does not take any passengers
- There are delays in AMS getting a gate assigned or a jetway connected to the aircraft
- There are delays in AMS in the handling of your luggage
- If you have to go through security in AMS, there may be huge queues at security
This is especially true on days when more people are travelling: week-ends (especially long week-ends), public holidays, beginning of school holidays (in the UK or in the Netherlands), etc.
The bad news is that you may thus be either delayed (possibly overnight or longer) or see your trip cancelled altogether.
The good news is that with the two flights on the same ticket, KLM is supposed to take care of you: rebook you on the next available flight, book and pay for your hotel if you have to stay overnight, give you meal vouchers, etc.
Of course since when this happens they are completely overwhelmed, this may take quite a bit of time to sort out.
- Check if your travel date falls on a week-end, long week-end, public holiday or the beginning of school holidays (in BOTH the UK and the Netherlands).
- If it does, it would probably be a good idea if you could move to another date which does not. If there is a change fee to do that, you may try to call them to have the flight changed for free, I'm quite certain they would prefer it a LOT if you travel on a less busy day.
- If you need to be in Lima for a specific event/at a specific time, you may want to move travel a bit earlier (a day or two).
- It may be helpful if you change the flights to a different route, e.g. via Paris rather than Amsterdam (KLM and Air France are the same group, and until now CDG does not seem to be suffering the same kind of meltdowns AMS has had).
- Alternatively, you may want to change the first flight to the earlier flight from MAN so there's more buffer in AMS (but the earlier flight from MAN is quite early).
- Make sure KLM have correct contact details for you (phone and e-mail) and make sure you check those regularly (including your e-mail's junk folder). They may contact you about cancellations, schedule changes, special instructions...
- Make sure you can read that e-mail on your phone
- Make sure you have contact info for KLM at hand
- Make sure your phone is fully charged before you leave, that you have at least a charger with you, ideally an extra battery (make sure it's charged)
- Make sure you have the KLM app and are properly registered.
- Check-in as early as possible for both flights (online or via the app).
- Follow the news for any disruption. Subscribe to the Twitter feeds of KLM and the two airports.
- If possible, know your options before you call. Try to find out viable alternative routings from both MAN and AMS (i.e. by order of preference, on KLM, Air France, Delta, and other SkyTeam partners). Try to find out the requirements in terms of passport/visa for alternative transits (don't ask to be rerouted via the US if you need a US visa to enter the US and you don't have one).
- Budget extra time to arrive early in MAN. Note that some airports give conflicting advice, as in "do not arrive too early as this adds to the congestion".
- Be ready when needed. You don't want to be the person who has stood in line for 2 hours and then takes 10 minutes to get ready to go through security.
- Make sure you have in your cabin luggage whatever is necessary for a longer-than-planned trip without access to your hold luggage, especially any medication, but ideally also whatever you need if you have to spend a night in AMS. Likewise you may get to your destination but your hold luggage may be delayed.
- Keep calm and be patient. Making a fuss will not change anything in the right direction in this situation. Being courteous and smiling may get you preferential treatment.
Remember that in the vast majority of cases, you won't face any of those issues, especially if you travel outside of the peaks. Trying to avoid them in the first place is probably the best strategy of all. It's also possible that by the end of the month both airports will have sorted out all their issues, but I wouldn't hold my breath for that.