Leveraged buyouts, as favoured by PE groups like Terra Firma essentially reduce the taxman's take by replacing the amount of corporation tax a company pays with debt interest used to fund the takeover. E.g.
Company A makes £1bn profit, pays say, £200m in corporation tax
PE Group X has £4bn and borrows £10bn to buy company A, transfers debt to company A
Company A now an extra £10bn in debt, pays £500m a year extra in interest.
Company A now only makes £500m profit a year, pays £100m corporation tax.
PE Group sells Company A for £8b, still makes a £4bn profit because debt remains on Company A.
So while there's not a direct link between PE activity and increased taxes for us there is a link between PE activity and reduced tax take from corporations, which could well mean increased taxes for us