Entry and exit stamps in Schengen-UK-Schengen

Since you're arriving in Madrid from a non-Schengen origin (Lima) and leaving on a flight to a non-Schengen destination (London), it's quite possible that you won't need to leave the international transit zone in Madrid. If that is the case, you won't be entering the Schengen zone, and not get any stamps, entry or exits.

Whether you need to leave the transit zone depends on whether you have luggage that is not checked all the way through to Stockholm. If it isn't and you have to collect and re-check it, you will need to exit the transit area and get stamps. But since both Otherwise you can stay in the transit area -- even if you need to change terminals there seems to be a non-Schengen airside transfer bus you can take.

The situation is similar for London, assuming you don't have to transfer between airports there. If you avoid having to collect luggage you can stay airside and won't have to pass through UK immigration. (Again there's an airside transfer bus between the various Heathrow terminals if you need it).

Since your entire itinerary is within the Oneworld alliance, presumably it is all on a single booking, in which case your baggage should be checked through all the way to Stockholm as a matter of course, and the first immigration point you need to clear is in Stockholm.

If you have separate bookings for some of the legs, things might get hairier.

Summary: It's not possible to avoid getting an exit stamp if you got an entry stamp but in this case, you will probably get no stamp at all in Madrid.

Third-country citizens (i.e. people who are not either EU/EEA or Swiss citizens) usually get a stamp every time they enter or leave the Schengen area. So if your partner gets an entry stamp in Madrid and leaves for the UK, then he should definitely also get an exit stamp.

If you would instead enter Spain and then travel directly to Stockholm from Madrid, he would only get an entry stamp in Spain and no extra stamp on the way to Sweden. But if you leave the Schengen area for the UK, Romania or any other non-Schengen country, then he would get a Schengen exit stamp. Whichever is the first EU or European country, the fact that you the UK is also in the EU or that you will return shortly are not particularly relevant, an exit is an exit.

There is no way around that and the only exceptions mentioned in the relevant regulation are holders of a residence permit for family members of EU citizens and people with specific travel documents like foreign heads of state, pilots, seamen, etc. In “exceptional cases”, the stamps can also be affixed to a separate piece of paper. None of that applies here.

But for a brief layover between two flights in or out of the Schengen area, he probably won't get an entry stamp (although if you want to see the city or leave the airport sterile transit area for any reason, then he would need one). So if we are talking about connecting flights booked together, he will probably not get any stamp before entering Sweden.