Now that the £9,000 cap has been lifted, fees here in the UK could rise substantially in the coming years.
Yes and no.
For the top tier of institutions, the fees will indeed have to rise considerably, to make up for the income lost from turning away international students to meet immigration targets, decreasing government funding, decreasing government research funding, loss of EU funding (research and infrastructure) and indeed just to help pay for staff visa costs. [A very significant fraction of research and academic staff are foreign, and this is absolutely essential to maintain - without knowledge exchange between international research groups, you cannot stay at the forefront of research. Visa costs now run at up to 1k per year per staff member.]
At the same time, the government is effectively creating a multi-tier university system via the teaching excellence framework (TEF). Universities in the lower tiers will have their fees frozen, which in real terms means they will decrease. (9k per year in 2016 is already worth far less than it was in 2012.)
BTW if you are funding university via student loans the increase in tuition fees does not make that much difference in the short run - students will still pay the same fraction of their income back, but the tuition fee increase will mean they have to pay for longer/be even less likely to pay it all back. The real issue for many families is the freeze in maintenance loan amounts at a time when student living costs are increasing very rapidly, as the missing money for maintenance needs to be found by families. This is a much bigger problem than tuition fee increases.