I was on a very limited budget for my first two trips to Europe and hardly paid any admission fees. I wandered around outdoors, soaking up the atmosphere. Forty years later, I still have not been up the Eiffel Tower or inside the Colosseum. You can certainly have a fine time without paying to go inside traditional sights. As already mentioned, London has world-class museums that are free (though a donation is requested), and churches are usually free as well (not always). Inexpensive food options are normally available at street markets, small supermarkets, bakeries, etc. So food, too, need not be a major expense. I suspect alcoholic beverages would be pricey, though.
The key factors will be the cost of airfare and lodgings. For the Amalfi Coast you'd have the additional expense of transportation from (probably) the Rome airport. Flying into Rome is usually more expensive than flying into the UK, so I'd suggest leaving Italy until you can spend more time there.
I agree with the others who've pointed out that you're talking about a very, very short trip. If the limiting factor is finances, I'd urge you to scratch around for enough money to stay longer. The flight days are a total loss, whether you have a 5-day trip, a 10-day trip or a 15-day trip.
Lodging costs at some destinations vary quite a bit by season. You must also be sure you're not trying to visit a city during a major local festival, because that will certainly drive up the cost of hotels; often you will find all the budget options already booked up.
Edited to add: It's pretty difficult to estimate someone else's food expenses, especially since you're now thinking of restaurants rather than cheaper options. Sodas can be expensive in Europe, and refills are virtually never free. In Italy it is conventional to order (and pay for) bottled water. It's a very rare day when I eat two restaurant meals; that would be too much food for me, and it would chew up more sightseeing time than I'd like. My one restaurant meal in an inexpensive spot (but not a chain if I can avoid it) usually costs between $20 and $30 (US), but it has occasionally gotten up around $40. My beverage of choice is water--tap water in places where it's not culturally frowned-upon to request it. Most meals are two courses because that's enough food for me. Three courses will usually send the cost north of $25.
When you settle on where you'd like to go, folks may be able to give you more specific information on food options. Italy is awash in great spots for pizza, which is usually about the least expensive hot, freshly-prepared meal you can get.