how undervalued are you in your current role?
a full time role paid at £15kpa should be very functional and task focused with little scope for demonstrating judgement or taking responsibility. there would be set procedures to follow.
a £45k role would require a very different skill set - it's not about working harder or doing more to get the higher salary. the role would have a lot more autonomy, judgement and prioritisation skills and probably some management too - very few people whose existing role is like the above will have had opportunities to demonstrate these skills professionally and consistently.
jumping to a £135kpa role from a £45kpa role is a different kind of leap. jobs are only paid at that kind of level for a limited list of reasons - the qualifications, experience and talent needed to do the role must be so rare that there is huge rivalry among competing employers to secure the people they need from a limited pool of potential. the job is likely to have so much responsibility that you would be expected to resign and maybe not work in the field again if something went catastrophically wrong on your watch, even if the mistake was made by one of your underlings. you would be expected to know everything that is going on under your authority, balance tricky competing goals and navigate complex issues, steering the activities of numerous people and avoiding risk factors that could lead to failure.
If you are being massively underpaid in your current role and you are being paid like a much more junior role while performing at a level closer to a more senior role, then you're in with a chance - so maybe just maybe if you are being paid £15k for a role that ought to be £23k then yes, or if being paid £40k for a role that ought to be £60k.
otherwise it would be hard to see how you could ever demonstrate the talents and experience needed for the next level in any recruitment process.