That's not acceptable, tbh.

Tomorrow when you call, and fail to get an appointment, I'd ask to speak to the practice manager. If s/he ain't accommodating, I'd ring your local trust.

I'm not saying this to freak you out, as the odds are heavily in your favour, but as an example, if my wife had kept being put off, she'd have been diagnosed too late for curative treatment. When her cancer was found and treated, it was on the cusp of being Stage 4. But she was seen the same day as she called by a nurse, not a GP. With the prompt appointment, it meant her cancer remained treatable.

GP surgeries are overwhelmed, that we all know. However, a worrisome lump takes preference over a sore finger. Don't let them keep delaying you. It's your health, and if you have genuine concerns, be your own advocate and push for an appointment. Badger them until you get to talk to someone who can actually get you in to see or speak to a nurse/gp

If getting through by phone isn't working, could you either e-mail the surgery or post a letter through tonight (marked urgent) explaining why you feel you need an urgent appointment (so mention new lump and relevant family history) and asking them to set one by for you. If there is something in writing it might make them think twice as you then have proof that you have requested to be seen urgently.

Make sure you keep trying to call in the morning as well... Hope you get an appointment soon!

I had this issue. 

I did an e consult and was advised to book a "routine" appointment! Same as you, when I finally got through all the appointments had gone.

So I resorted to going down to the gp and sitting in the waiting room refusing to move. I now have an appointment on Monday.

I know that isn't a great thing go do but same as you I've found a lump and am scared to death!! 

Totally unacceptable. 

I just can't fathom a scenario when a lump in your breast is not considered urgent for an appointment.

I know all surgeries will be different but when I had breast symptom (nipple discharge rather than a lump) I called my GP at 4.00pm had a call back within half hour and was seen before they closed at 6.00pm the same day.

I really hope you get an appointment soon and perhaps go with the advice of previous reply and turn up at the surgery. 

Take care x

I'm an ex GP with cancer. The GPs will want to see you urgently. The vast majority of breast lumps are nothing to worry about but you need seeing and almost certainly a referral for a scan.

Most GP surgeries have ecconsult usually seen same day by GP. If you can speak to receptionist say you think you have breast cancer and need a same day appointment. If she refuses ask to speak to GP or practice manager. 

It's probably nothing to worry about but you need to know that for sure,  especially with your family history.

When you ring ask them to put your name on the triage list , say what you are worried about and a dr should ring you back usually within the hour they can decide if they want to examine you or just refer you to the Breast clinic for a scan, l hope you have more success making the appointment. 
take care 

Susie