We aren't full due to space but we don't have the infrastructure/resources for the additional people. Some of that is due to political parties not investing when they should but it is also difficult to keep up and expensive to make major infrastructure changes. You also can't magic doctors, nurses, teachers out of thin air. They need to be trained which is a long process. You can build a school or hospital far quicker than you can train enough people to staff it.
Population growth is fine as long as infrastructure growth is the same speed but it hasn't been. The population has increased 20% over the last 30 years. Infrastructure has been nowhere near that and in many places it has gone backwards.
We do need more infrastructure, but just to the level we should have had already etc, the infrastructure didn't grow at the same rate as the birth rate required it to, and now we're putting sticking plasters on gaping holes. i.e HS2 for example should have been built when the channel tunnel was. I think it's going to be difficult to get parties to now commit to massive infrastructure projects now, when they know the population will likely start to decline in a couple of decades. It's like we've missed the boat on that, but there may be tech advancements where we can invest in new things, and sort of skip a generation.
The population has grown, but it's also grown much older, in the 50's the median age was 33, now it's 40, and by 2100 it's going to be 50. Some of that is due to living longer, but we're not really gaining in length of production time compared to what people are gaining in length of less productive time. So, although we have "more" people, I bet we're not gaining in activity at the same rate. We're going to need to invest in care homes, that's for sure, we're going to run out of those quickly.
We need to be much more services focused than we already are, and have more people working from home etc, it will help on the burden we have on existing infrastructure. It's also what we're best at, and chasing work in areas we're worse or less competitive with the world at, doesn't make sense. We currently can't seem to afford to maintain our current infrastructure, the state of the roads is a good example of this.
Affording anything is going to be the problem, especially for the future, we need more younger folk now, as in 20 years we're going to need them when they're 40-50, at peak output, peak tax etc. We're also going to have to massively change taxes and get wealth redistributed, or we're going to go further into the hole.