Hello Amy,
I am a few years older than your Mum, so I am very sorry to hear of her secondary cancer, that must add hugely to your anxiety just now.
You asked what happens at a breast clinic, so here goes. The hospital where I live has a 'one stop clinic which means that you leave with a diagnosis, in some places you may have to go back for some of the tests or for results, so it may be worth checking how your hospital works. The overall procedure is pretty similar, and will stop once they don't need to test any further.
To start with you will see a doctor who will ask about your lumps and then will examine you. Because of your age, and because you are pregnant, they probably will not do a mammogram but instead do an ultrasound to see what they can see. This would show whether the lumps are solid or if they cysts. If they find cysts, they may well drain them and hopefully that is that.
If they find solid lumps, they may well do a fine needle apsiration (FNA) or core biopsy to find out what kind of lump it is. They numb the breast with a local anaesthetic and then use a long thin needle for the FNA or a special tool for the core biopsy. They will either give you pain killers to take home or tell you to use your own painkillers, as once the anaesthetic wears off it can be a bit sore for a few days. The results from a FNA can be in about an hour, and they will then have a pretty good idea what the lump is. Hopefully you will be in the 90% of women whose lump is benign and you can go home relieved.
The results form a core biopsy are more precise, but take a few days to complete all the tests. This would confirm a diagnosis and allow the doctors to decide the best treatment plan. If you are unlucky enough to have cancer, they will take into account your age and your pregnancy in amking those decisions with you. There other young women on here who have been diagnosed during or just after pregnancy, and I am sure that they will be able to share good information with you.
The waiting and not knowing is the worst part of this whole thing, so try to keep yourself busy this weekend, doing nice things. If possible take someone with you to the breast clinic as it does help to have someone else to pick up the things you might miss - especially if it turns out to be the news you dread. Hopefully it will be good news, and they can buy you coffee (or something stronger!) afterwards.
Keep posting as you need/want to, there are no silly questions. You might also like to give the BCC helpline a ring, as they are very helpful and supportive.
Gentle hugs.