I am about to install an oil fired combi boiler (15/19), I understand that this will allow me to get rid of my immersion tank as the 'combi' will give me hot water on demand, I am concerned that it will not cope/ keep up with bath tap running... will the water keep running hot or will it go tepid. I have 3 young kids so it's quite important! Many thanks.

What is the model of the boiler?
How much is it rated for in output for domestic hot water per hour?

With the information you posted there is no way to know.