I have actually done this (and you can search for the thread I started on the subject.)
My baby was almost a month old when we travelled. I'd had several children already, and a track record of fast, uncomplicated births.
My concern here is not the baby so much (all those issues have already been covered by previous posters) but YOU. As a first time mother you have NO IDEA what the birth will be like, how it will leave you physically, psychologically... There is no way I'd have been able to handle a trip abroad with my first at a month old. Be prepared to not feel able to manage it - and prepare your family too, so that they don't pressure you, or get too upset if you can't make it after all.
Practicalities: you cannot get a passport (in England) without a birth certificate. These are issued by your local council; check their website, and ring them and ask what likely waiting times for an appointment will be in July/August. Could be a week, or it could be a month.
The fastest passport service offered for children is the 1-week Fast Track; you book an appointment, to which you must bring the birth certificate (see point above). These appointments are often booked up for weeks in advance. Again - check your local passport office, and be prepared to travel to the other end of the country for a sooner appt if necessary.
Once you've got your birth certificate and been to your appointment, your baby's passport will be delivered to within a week - but everything depends on how quickly you've managed to register the birth and get a Fast Track appointment.
You can buy suncreams specifically designed for babies, but in the end I just always kept her in the shade, and bought a UV umbrella for walking from A to B.
The baby will almost certainly be fine (as mine was); it's you that's more likely to have serious issues, all of which will be far easier and more pleasant to deal with in your own home.
Good luck!