The train takes a day and a half including a night on a sleeper train, so chances are you won’t get any sleep and will be tired for a few days after arriving in Portugal. Flying is much easier.
All airlines have hand luggage restrictions, some more than others eg EasyJet just have size restrictions and no weight limit, whereas others have different size restrictions plus weight limits. Many of the value airlines use Gatwick, Luton, Southend and Stanstead, not Heathrow.
There is lots to see in London- you don’t say what your interests are, but most want to see the Houses of Parliament, Buckingham Palace, some of the museums and galleries and the Tower of London.
There are plenty of accommodation choices in the centre of London, from hotels to apartments - anywhere in tube Zone 1 is suitable. A good option is often the Premier Inn chain, although the best value rooms will have been taken for June by now. You don’t say what your budget is (in GBP).