Hi, I'm doing economics & finance here. First of all, don't worry about the programme, recruiters don't discriminate between BIEF and BIEM, contents of the first year are the same, with the difference that at management you'll have higher grades (useful for applications, exchange programmes at prestigious unis, admission to MSc).
Bocconi undergrads get into spring weeks, summers, etc. Going into IB straight after the undergraduate degree is definitely possible and has been done. However, in most cases, IB hires have a Masters here because it gives you more time to get valuable work experience, acquire technical competence for interviews, explore other career opportunities, etc. Furthermore, with an MSc you are sure you'll never have to go back to school, you are actually more qualified (this is taken into consideration more for jobs in Milan than London) and promotions will come naturally. In Italy with a 3 years degree adults and oldies won't even consider you an actual "Dottore", but in the UK they care much less. If you are 100% committed to IB since day 1 and have enough work experience on your CV to apply to SWs and summer internships, then you don't necessarily need a master.
This said, due to (hard) Brexit it's becoming more difficult for non-UK students to get into London. If the City stops being Europe's financial hub for investment banks, as an Italian you will hardly find open positions there. But in that case, it would also lose much of its attractiveness.
Finally, as a rule of thumb, if you want to work in London go to a London uni, you will have more chances. I would be uncertain between UCL and Bocconi, but would definitely go to LSE if I had the chance!