The NHS is actually pretty efficient when it comes to the two week referral for breast cancer, in my recent experience. So the likely timeline is, if you contact your GP, you will be referred for scans/biopsies at a breast clinic within two weeks; it could well be sooner.
If the clinic finds anything suspicious, they will tell you on the day, and if they are fairly sure it is cancer, they will book you an appointment with a consultant, which should be within another two weeks.
If surgery is needed, it should be around four weeks after that. So in total, you are looking at around eight weeks from first contact to surgery, if needed. This may turn out to clash with your holiday, if it is early June, so you will then need to decide if you want to cancel, or to wait an extra week or two for surgery (ask advice from the doctor about this). You are unlikely to need emergency medical treatment for a breast lump while on holiday.
I think if you haven't taken out travel insurance already, and you are aware of the potential issue when you buy a policy, the insurers would be justified in refusing to pay out for cancellation - be prepared for questions about when you first noticed symptoms. This is why you should always buy insurance at the same time as you book a holiday.
Of course, the chances are that it will be something benign like a cyst, and you will be able to go on holiday, relax and forget about it.